Spock in a volcano. Not only does this allude to Vulcan, his race named after the Roman god of fire, but also symbolizes his calm and stoic appearance that hides so much emotion that can just burst out.
Harrison's regenerative blood isn't as much of a gamebreaker as it was in, say, Heroes, since in the Trek verse most deaths result in total bodily disintegration anyway (or being eaten by rock monsters, melted into goo, exploding, imploding, etc.). But back in the time Harrison was created for (300 hundred years ago from the 23rd century, according to the movie, putting his glory days in the 20th century this would have stymied a lot of people.
The true villain's plan: The Admiral ordered the Enterprise to fire on the Klingon homeworld, but sabotaged it so it would be stranded at the edge of Klingon space. He then intended the Klingons to attack it, granting him the causus belli he needed to engage them back in his own dreadnought which "coincidentally" would arrive just in time. Indeed, the Admiral has attempted to replicate the Kobayashi Maru scenario.
The only person who refers to Khan by his full name is Old Spock, who never met him face to face. John Harrison is a code name.
It may seem confusing that the main message is not to seek vengeance when most of the movie is all about vengeance, particular with Kirk's (and later Spock's) fight with Harrison. But it makes perfect sense when you realized what would have happened if they hadn't stopped their No-Holds-Barred Beatdown. If Kirk had killed Harrison right there when Harrison surrendered, he would not have learned of the fact that Admiral Marcus was conspiring to start a war with the Klingons by sacrificing the crew of the Enterprise. Likewise, if Kirk had not followed Spock's advice and simply went along with Marcus' plan to launch stealth torpedo to kill Harrison as opposed to arresting him, then the Enterprise would have provoked an unintentional attack on the Klingon's home territory, causing an outbreak of war between the Federation and the Klingons just as Admiral Marcus had planned. And if Spock had proceeded to kill Harrison to fulfill his vengeance after Kirk's death, then he would have lost the only chance of saving Kirk's life.
The running theme of the film is how far would you go for your family. Harrison, Kirk, and Spock go to great lengths to ensure the survival of their crews because in the most desperate times, when you've practically lost everything, your crew is the only family you have left and you will go to any length for them, even murder, Heroic Sacrifice, or genocide. Harewood, the officer who committed the suicide bombing for Harrison, only committed the act if it meant that his daughter would be cured of her illness.
Related to the above, perhaps this could also be true for Admiral Marcus as well. He saves his daughter from what he's gonna do to the Enterprise and he may have been thinking that his militarisation of Section 31 and Starfleet would only be in the best interests of humanity, another type of family, just as Harrison considers all of his fellow super-humans (most likely regardless of any actual blood-ties) his family. Even Keenser and Scotty have a family dynamic.
Why would the prime-universe Carol Marcus have defended Starfleet from accusations against it from her own son? Because her own father was probably a Starfleet officer.
Harrison describes the Vengeance as being "dreadnought-class" while naval tradition holds that the class is named after the first vessel commissioned. (E.g. the U.S.S. Constitution would be the first Constitution-class starship). Marcus was planning to fight a war with the Klingons and you can't fight a war with just one ship, no matter how powerful it is. Thus, the U.S.S. Vengeance is not the only ship of her class out there, it may have simply been the one Marcus could get to the fastest.
Why is Khan white when he was described as Indian in "Space Seed"? Well, considering how being a genius is a requirement to take just the entrance exam to Starfleet, they couldn't risk having a 20th Century criminal walking around without someone recognizing him, hence major plastic surgery and a race lift.
Khan has 72 members in his "family." This number is significant because it's the same amount of disciples Confucius had, the evil Egyptian god Seth had, and even Jesus had according to some accounts. To different people, Khan can be Confucius, Seth or Jesus. According to J.J. Abrams, with the numerous parallels to terrorism within the film, both intentional and unintentional, they themselves worried that people might draw comparisons between the 72 missiles and 72 virgins, after noticing the coincidence in numbers.
In Space Seed, Khan mentions that he was an engineer (although he uses this as an excuse to get his hands on the ship's technical specs). If we take him at his word, though, this helps explain how he'd manage to design and build a next-gen warship.
The first shot we get of Qo'noS shows a half-exploded moon. In the original timeline, Praxis exploded because of over-mining, resulting in a catastrophe in 2293. But then the Narada and Nero arrived in 2233, and we're told that the "Kelvin incident" prompted the acceleration of Starfleet's militarization and technological development. However, the Klingon Empire, which ultimately captured the Narada, held it (and presumably studied it) for 25 years, and then suffered the loss of 47 ships when Nero escaped, was also affected. The Empire's own changed timeline and rising tensions with the Federation possibly accelerated the mining of Praxis, and, in turn, the catastrophe. Which might also explain why the Klingon city that Khan was hiding in was devastated and abandoned. Even more so, the Narada was a mining ship.
Why are the Caitian twins that Kirk sleeps with furless? They'reshaved!
Spock's shout of KHAAAAAAN!. Is it Narm? Sure. It's also an emotional reaction from someone who has spent all his life repressing emotion rather than learning normal ways to vent it. In the past, this kind of emotional breakdown has led to a No-Holds-Barred Beatdown of the person who triggered it. That's impossible in this case, since Khan is not physically present. His human colleagues might have cursed, yelled illogical insults about Khan's parentage, made overblown threats, punched walls, etc.—coping techniques that Spock has never learned. No wonder his reaction is a little Narm-y—the poor guy has no better tools for dealing with this once his emotional control fails.
Why does not one, not two, but three Kilngon ships corner Kirk and co. on Kronos when they were positive they'd be fine in what is an uninhabited area? Because the Klingons are already searching for "Harrison." When Uhura explains what the crew is doing by saying they are hunting a "criminal," she doesn't identify his species, crime, or anything beyond the fact he has killed humans. It is the Klingon commander who identifies her target as human. The Klingons realized a human had teleported to their planet, but figured he was just a spy. It's why they want to capture the crew, and why there are so few of them.
The USS Vengeance is almost fully-automated, being designed to be operated by a single crewman if necessary. This actually proves to be an Achilles' Heel, as the small crew is unable to effectively detect and repel infiltrators (Scotty) or an enemy boarding party (K2), and makes it entirely possible for that same boarding party to sabotage and seize control of the entire ship within minutes. Although this makes a lot of sense, for a "pure warship." For a massively spacefaring culture who can mine uninhabited planets and asteroids, having the resources to build something is trivial: the real supply bottleneck is trained personnel to operate and maintain their machinery.
Bones describes Kirk's plan for Sulu as telling him to bluff. The rest of the film is Kirk, Khan, and Marcus all trying to play the other two against each other so they can forward their agenda without revealing their own— a giant poker game. (This is particularly apparent in the first scene with the Vengeance, where Kirk uses his superior knowledge to lie until he can figure out the disparities in Khan and Marcus' plans, pick out which bits appear most true, and protect the Enterprise. Kirk flawlessly plays Khan against Marcus, but underestimates Khan's hatred for Starfleet as a whole; luckily, Khan underestimates Spock's ability to bluff.
Khan is over 300 years old, and Marcus woke him up after the planet-destroying events of the first film. It's likely Khan has never met a Vulcan, much less a half-Vulcan, which is why he so dramatically overestimates Spock's inflexibility.
At least one reviewer criticized the space jump, asking why they didn't use a shuttle to get to a ship that did not have weapons. For starters, I don't believe the TOS shuttles were ever established to have transporters on them. Second, even if their weapons weren't working, their sensors probably were, so Admiral Marcus would have been able to see a shuttle taking off and prepared for a boarding party.
Why did John Harrison agree to save Lucille Harewood's life, in exchange for cooperation from her father? Harrison knows what it's like to not only lose family, but to be helpless to do anything for them.
Another Tear Jerker: Why did Spock agree to the volcano mission? He already saw his home planet destroyed and after evaluating all their possible options, his human side didn't want to see another race destroyed.
Halfway into the movie, Kirk sports a cut on his face that resembles the Starfleet symbol after his encounter with the Klingons and John Harrison. It's subtle Foreshadowing that Starfleet (or specifically a Starfleet admiral) was the one who manipulated Kirk and indirectly caused that injury to him.
The naming of the USS Vengeance. Aside from Rule of Cool, its name alludes to the theme of revenge throughout the film ( Khan wanting revenge on Admiral Marcus for killing his crew, Kirk wanting revenge on John Harrison for killing Pike, Spock wanting revenge on Khan for killing Kirk). Not to mention the main plan of Admiral Marcus of sabotaging the Enterprise and having them killed by Klingons to justify their reasons of avenging the Enterprise crew and going to war with the Klingons.
The second time our attention is directed to the new torpedoes, Scotty explains that firing them could start a chain reaction and potentially cause the Enterprise's warp core to detonate...but this is not the second time the audience has seen the torpedoes. You can see a prototype torpedo in the background as Harewood walks to his desk. Scotty's explanation is roughly the same way Harrison gets Harewood to destroy the Section 31 building: He drops his ring into a glass of water, which causes a relatively small explosion, and the scene cuts to a chain reaction of escalating detonations, presumably of the torpedoes. Harrison knew exactly what kind of explosion Harewood would have to create to cause the reactions, and may have even intentionally wiped out all (but a very specific seventy two) of the prototype torpedoes. Perhaps Harrison's need to manipulate someone in her father's position is the real reason Lucille Harewood is in the hospital in the first place...?
Kirk promised the safety of Khan's crew in exchange of helping them stop Admiral Marcus, to which Khan expressed incredulity at such a promise, since he believed Kirk couldn't even save his own crew. Kirk does keep his promise when Spock and McCoy removed Khan's crew from the torpedoes, kept them safe in the med bay and used the torpedoes to cripple the Vengeance to save the crew. Not to mention Kirk's Heroic Sacrifice.
In Kirk's death scene, look at the expression on Spock's face when Kirk tells him "This is what you would have done". It's small, but it shows understanding. Now think back to earlier on when Spock contacted Spock Prime and asked him about Khan; when Spock Prime explained that Khan was defeated "at a great cost", Spock asks how. It's very likely that Spock Prime told him about his Heroic Sacrifice, adding more hidden depth into the death scene.
Similar to above; the expression on Spock's face when Uhura offers to play diplomat to the Klingons. You know at that moment, he realizes what Uhura must be feeling whenever he goes into a life-threatening situation, only then fully understanding why she was so upset before.
Why is Spock so suspicious of Carol when she first presented her credentials? Because as the ship's XO, Spock has to double check and confirm all the necessary paperwork and credentials for anyone who boards the Enterprise before Kirk can sign off on them. Transfers would have gone through him and since Carol presented a last minute transfer, this raised a red flag with him and led him to further investigate how this got past him and the scene where he asks Carol why she falsified her credentials.
At first, Harrisingh and his crew getting refrozen at the end was a bit of a headscratcher for me until I remembered: he and the other augments are three hundred years old. The governments that tried them are long defunct if they were even given a trial in the first place. Organizing one in the Federation would be a prosecution nightmare. No eyewitnesses still alive, all the bodies turned to dust, not to mention that any pretense of innocence is out the window. Legally everyone but Harrisingh could be free from the minute they were unfrozen. As for Harrisingh, the whole crazy Admiral forcing him to wage a war on the Klingons could get a lot of people angry. Refreezing them was the best option.
In the first film, Amanda Grayson in their last moments looked at Spock in the eye, knowing that their loved ones would be saved. In the second film, Jim Kirk in their last moments looked at Spock in the eye, knowing that the Enterprise would be saved.
It doesn't make a lot of sense outside of plot device for Old Spock to say anything about Khan to Young Spock. Being a mostly logical person, he would have stuck to his previous statement and not revealed any information about anything that might change things for the worst. But when Young Spock says the name Khan, it brings up some simple, base emotions in Old Spock. Khan killed him (sort of) and he was just feeling a bit too much of that human side emotion and decided he did not want Khan to harm anyone else. Of course we all know how well that worked out. Which leads to another moment of Fridge Brilliance...
Khan really is as smart, cunning, and powerful as he is made out to be. No matter what anyone does - known past, alternate present, or unknown future - he still ends up one of the most dangerous people to ever have existed. Killing hundreds if not thousands (or more) all for the sake of whatever his personal cause might be. One must wonder why they keep him around.
Admiral Marcus has a whole bunch of models of famous air and space craft on his desk (seen around 30 minutes in), including the Wright Brothers plane, the Gemini capsule, Zephram Cochrane's warp ship... And the USS Vengeance, at which he's staring when he comments that war with the Klingons is inevitable. Where this turns into brilliance is that Marcus is engaging in Refuge in Audacity, keeping a model of his super-secret prototype warship in the open for anyone to see. How better to convince everyone it's actually a non-secret pet project that he has no real intention or capacity to build?
A meta example: When he was younger, Richardo Montalban (the original Khan), looked a lot like Benedict Cumberbatch if you discount the different skin tones.
The Vengeance and the Defiant:
The Vengence was constructed for pretty much the same reason as the Defiant of DS9 fame. Both were constructed because the Federation now had new, far more powerful foes that diplomacy wouldn't work against, so purpose-built warships (instead of multi-purpose ones, like the various incarnations of the Enterprise) were needed. And had this been a different universe where the chief designer wasn't an axe crazy General Ripper, the Vengence might have actually been a pretty cool ship and helped Starfleet quite a lot, just like the Defiant did.
Furthermore, it also makes sense that the Vengence is so huge and obviously warlike compared to the Defiant, which is classified as an "Escort Ship." Vengence was built in the aftermath of the destruction of Vulcan, with the history of the Federation being far shorter and more turbulent (the Xindi attacks, Temporal Cold War, the Romulan War, various conflicts with the Klingons, etc). Them having the will to build something like the Vengence makes more sense. But by the time of TNG, the Federation has been in a period of relative peace and stability. They claim to have moved "Beyond" such primitive things as "revenge" and they accept death. While they get shaken up by the destruction of the fleet at Wolf 359 by the Borg, the Defiant is still very small and almost cancelled, which makes a lot of sense if you consider cultural inertia. People wouldn't want a warship when all they've known is a peaceful starfleet, and it takes the Dominion to shake them out of their stupor and start building more straight-up warships. The younger Federation has no cultural inertia, so it is more capable of building these vessels with fewer moral hangups about it.
The Tear Jerker scene at engineering Kirk doesn't believe in no-win scenarios and must have mustered all his strength to get out thinking that he can still survive only to find the door locked. He then closes the other door, probably hoping that it will block the radiation but then he realized he's dying and told Spock, "I'm scared..."
There's a live tribble on the Enterprise. A live tribble that has been enhanced by John Harrison's super-blood, which brought it back from being dead.
The USS Vengeance crashed into the city of San Francisco, and the broken remains of the ship are now sitting a few blocks away from Fisherman's Wharf. What about the Warp Core? That great big can of anti-matter capable of powering such a ship? I don't envy the engineers with the responsibility of having to make-safe the wreckage. Of course, since the computer told Khan that there was insufficient power, it's possible that he had to eject the warp core off-screen after Spock's Trojan Horse attack.
Mentioned above, the first ship of a class is usually named after said class. (Constitution Class = USS Constitution, Intrepid Class = USS Intrepid, ETC). There could literally be a dozen other Dreadnought Class ships out there and we and Starfleet have not idea where or how many. The most advance warship of the Federation, hundreds of governments and criminals angry at the Federation and it only takes one to pilot it. Furthermore, who knows how many other Dreadnoughts are out there under the custody of Section 31? Nobody in Starfleet has any legitimate accountability of these ships, and who knows what the men formerly under Admiral Marcus's command might choose to do with such a force, either against Starfleet, or even worse, on their behalf, against the Klingons or Romulans or Gorn or some other group. They could start a massive interstellar war, with the Federation and Starfleet left to answer for it.
Aside from the obvious moral implications of killing people without a trial (or with one, the Trek-verse humanity seems to be against capital punishment), they left all 72 of Khan's chosen - and Khan himself - alive and in stasis in what appeared to be some kind of secret government warehouse (maybe parked near a certain Ark of a certain Covenant?). The horror is why they are keeping them someplace where they can find them? Protection from being "discovered" and let loose? Or perhaps something more sinister.
Sulu's transmission to Harrison demanding his surrender. He identifies himself as a Starfleet officer, states that other Starfleet officers will be landing on the Klingon homeworld, and threatens to bombard the planet if Harrison doesn't surrender himself. On a presumably public channel, where the Klingons are sure to hear it. Sulu just gave the Klingons the perfect case they need to start a war, as his threats and Kirk's actions could reasonably be considered acts of war. At the best, Sulu could easily be justifiably extradited to Qo'noS on that statement alone.