Montalban could not film along with the main cast because of his Fantasy Island commitment and this prevented a planned Khan/Kirk in person fight. Hence, the two only interacted via communications and ship to ship battles which actually created a far more realistic encounter.
From this movie onward, everyone wears red uniforms. By virtue of everyone wearing a Red Shirt, that means the meta-possibility that Anyone Can Die. Sure enough...
For Kirk's birthday, Spock gives him an antique book. Bones gives him reading glasses.
Bones insists that Kirk needs to take command of a starship, lest he waste away behind a desk. Spock insists on handing over command of Enterprise to Kirk.
Thanks to Star Trek (2009), we now know why Spock never took the Kobayashi Maru test. He designed it.
When Reliant is first hit, they can't fire the torpedos or the phasers, despite only having the warp drive and torpedo controls damaged. But in the first movie, Decker says that the phaser power is channeled through the warp drive... meaning that Reliant can't fire back!
According to Decker, the ship's "new" design (ie, part of the Enterprise's recent refit) channeled phaser power through the warp core to increase firing power. However, the Reliant is a Miranda-class starship while the Enterprise is Constitution-class. Ships in the Star Trek universe have a service life of 30+ years so these vessels could have been developed at different times with unique features and capabilities.
True, but when Enterprise received her refit, she received some very distinctive design features that she previously lacked, such as the warp nacelles lacking the glass domes of the original design. Reliant shares many of these features, suggesting that she shares many of Enterprise's engineering choices.
At first it seems strange that Khan's crew is so young, far too young to be his original crew, too old to be his crew's children unless Augments age faster than regular humans. But then you realize that Khan's crew parallels Kirk's—both ships are being staffed by "children" who were not involved in Kirk and Khan's original conflict, but who are facing the ramifications now.
A related parallel between Kirk and Khan. Kirk gains a son who has been estranged from him for his whole life. Khan loses a son-figure (Joachim) during his quest for vengeance.
And, Khan was the leader, whose pod was set to revive him first. So he possibly spent more time outside his pod than the others in the group.
In the semi-canon novel To Reign in Hell Khan's crew are indeed the offspring of Khan's original companions who all died in the years after settling Ceti Alpha V. Joachim is the son of Joaquin (Khan's faithful right-hand man in "Space Seed") and was adopted by Khan after his father was killed defending Khan from mutinous Augments.
Kirk's Say My Name shout of "KHAAAAN!" has become the go-to example of what a Large Ham he was... but in this case, he was invoking it on purpose. His plan depended on convincing Khan that he'd won, so he plays into Khan's pettiness and vanity by giving the most over-the-top scream of despair and rage that he can. As soon as Khan is no longer listening in, he calms down completely.
Before Terrell turns his phaser on himself, the last command that Khan gave him was, "Kill Him, Terrell!" neglecting to mention Kirk by name in that sentence. So Terrell made his final sacrificial act of defiance by still actually obeying Khan's command, except literally, as if Khan meant, "Kill him (Terrell)."
It's mentioned in several places around this Wiki that Bones's gift of reading glasses to Kirk, due to an allergy to the only medication that can treat his eyesight, is an example of Science Marches On made obsolete by the invention of Lasik surgery. However, Lasik (and other, earlier surgeries) treat refractive error in the eye. Age-related presbyopia, which is the most common cause of needing reading glasses as we get older, is caused by the eye's lens losing flexibility with age, preventing the eye from changing focus to different distances. That's something which could quite plausibly be treated with medication rather than surgery, and science hasn't marched on yet: although it's plausible that a flexible artificial lens could be invented, it hasn't been yet.
The ship that transported Khan to Ceti Alpha V was named the Botany Bay, presumably after the real-life geographic feature at what is now Sydney, Australia. Being "sent to Botany Bay" is old British slang for being "transported". Essentially, the Federation did this exact thing to Khan and his followers, and might have chosen a ship of that name just to further sting his pride.
Except that the Botany Bay was the ship Khan and his followers were on when the Federation found them. That their journey aboard the ship ended essentially with them at a penal colony was just an interesting bit of irony.
Khan and his followers were exiled from Earth in suspended animation. The name is almost certainly deliberate, but not as a reference to Ceti Alpha V; rather, it was referring to their initial exile.
Much is made of how simple Spock and Kirk's verbal code is while they're talking repairs, and how Khan should have seen right through it. However, the code relies on a piece of knowledge Khan couldn't have had: namely, that Saavik is very fond of quoting and following Starfleet regulations. Recall Spock's inflection: "If we go by the book, like Lieutenant Saavik, hours would seem like days." The latter is letting Kirk know what the code will be, but the first two-thirds is letting Kirk know they need to speak in code in the first place. Without knowing who Saavik is, her fancy for quoting regulations, or that there's a Starfleet regulation about "no uncoded transmissions on an open channel," Khan couldn't decipher the code because he had no reason to suspect there was a code in the first place!
At face value, it even sounds like Spock is describing Saavik as a rigidly unflexible officer who can't adapt to the problem, and that the original estimate given is their "cutting corners and rushing" estimate. The fact that Spock actually holds Saavik in high regard is something that Saavik and Kirk would know, but Khan could not. For her part, Saavik, while indeed quite fond of pointing out when Kirk is neglecting a regulation, is more than flexible enough to follow the lead of her seniors despite her concerns, even if it does occasionally lead to disaster, as with Kirk's failure to raise shields earlier.
When the Enterprise fires the final three shots that disable Reliant, they're all non-lethal shots. The first shot takes out the Reliant's torpedo bay, the second shot disables Reliant's port warp engine, and the final shot knocks the engine off entirely (meaning that if the antimatter in the warp core goes kablooey, it should be too far away to do any damage to either ship). In other words, Enterprise is making sure Reliant can't use her weapons; the torpedoes are gone, and she can't fire phasers without both warp engines. At this point, even though Reliant is very badly damaged, she's still salvageable. After all, a Starfleet ship is an expensive piece of hardware, so why blow it up outright when they can tow it in and start to make repairs? Unfortunately, Khan had one last trick up his sleeve....
The warp core isn't in the nacelle, it's in the Engineering section in the main part of the ship. Warp nacelles are just big sets of field coils; they're powered by bursts of high-energy plasma delivered via conduits from the warp core, and when energized, they create the warp field that allows the ship to travel faster than light. The part we see in Engineering is where pulsed amounts of matter are mixed with similar amounts of antimatter (the vertical shaft), and the resulting plasma travels down the horizontal shaft towards the back of the ship, until it splits into two conduits that go up the pylons to be delivered to the nacelles. Thing is, that blue dome on top of the impulse engine that gets shot out in the first duel is actually the top of the matter/antimatter core, at least according to the old Mr Scott's Guide to the Enterprise, and presuming Reliant is similar, that's why Joachim says they can't fire because the warp drive is damaged. As phaser power is (normally) channelled through the warp engine, they can only fire torpedoes. Scotty managed to give Kirk a few shots via the backup batteries but Khan never manages to get his phasers working again. The nacelle being shot off means Reliant can't go to warp (if it even could), but it would probably also cause a lot of internal damage from plasma and coolant conduits being ruptured too.
Khan seems pretty put out that Terrell didn't know about him, that Chekov didn't even think to tell stories of him to entertain his captain during their voyages. If Chekov didn't think to mention him, it's probably because Khan was, prior to this film, not a particularly memorable character by the standards of the Enterprise crew. The ship has met multiplesufficiently advanced aliens, including Apollo, of Greek mythology. Chekov was on the mission to intercept V'Ger, a sentient AI driving a mind-bogglingly massive starship, before it could destroy Earth. Khan, in comparison, is an unusually strong and clever human who one time failed to take control of the ship and absconded with one junior member of the crew twenty years ago. This of course is ultimately what makes him so dangerous: None of the Starfleet officers thought enough of him to expect him to ever become a threat again until it was too late.
Pay attention in the scene where Enterprise hacks into Reliant's computer and lowers her sheilds remotely: Spock enters the code from a station at the rear of the bridge, but when Khan is trying to figure out how to override the code on Reliant, he is looking at one of the consoles at the front of the bridge. As Michael Okuda noted in the text commentary for the scene: Khan isn't going to find the override on that panel.
The Shaky P.O.V. Cam rapidly panning back and forth across the console serves to sell Khan's unfamiliarity with Starfleet ships. As he suddenly realizes that his carefully-concocted plan is coming apart unexpectedly, he panics and is unable to regain the initiative before Enterprise cripples Reliant and limps away to relative safety.
The Kobayashi Maru simulation features a distress call from a ship in the Klingon Neutral Zone. The moment the Enterprise crosses into the Neutral Zone, however, the Kobayashi Maru disappears from the scope and a squadron of Klingon battlecruisers appears, ready for battle. The implication of the scenario is that the original distress call was a Pretext for War, with the Klingons luring a Starfleet ship into breaking the treaty so they could ambush it. In fact, this was basically the plot of the Next Generation episode "The Defector," but with Romulans luring the Enterprise across the border, and ironically enough, the Klingons helping the Federation to defuse the situation.
More to the point within this film's narrative, the distress call being bait to lure the Enterprise into an ambush is also essentially Khan's plan.
The film gives no indication of what happened to the Reliant's Starfleet crew; we can only guess from a limited number of very bad possibilities:
Khan left them for dead on Ceti Alpha V, a barren hellscape where they'll probably have no idea to find food, water, and a defense against the Ceti eel while they wait for potential—but uncertain—rescue (henceforth known as 'the optimistic scenario').
Khan tortured them for information before brutally murdering them, like he did with the scientists on Regula 1.
They were being held hostage on Reliant when it was destroyed. Some of them may have even been coerced into helping run the ship—coercion that probably involved more Ceti eels.
Option one is the actual scenario, as in the movie, Kirk's last log entry explicitly states that they're heading to Ceti Alpha V to pick up the crew of the Reliant. It doesn't seem like much more than a week or so passes (roughly according to the stardates in the movie), so it's not quite as bad as first thought for the Reliant crew.
Captain Terrell explains this earlier in the film, when the landing party finds Terrell and Chekov at the Space Lab.
Saavik: Where is the Reliant crew? Dead?
Captain Terrell: Marooned on Ceti Alpha V.
The novelization says that they used eels on the engineering crew (who thus perished), and left the rest marooned on Ceti Alpha V.
Arguably a bit of Fridge Brilliance, as this explains how Khan's crew knew how to bluff Enterprise by claiming their comms were out due to a Chambers Coil overload.
Did Kirk end up insulting Spock at the latter's own funeral? Granted, Spock got better, but consider the following:
Kirk: Of my friend I will only say this: of all the souls I have encountered in my travels, his was the most ... human.
Kirk: Spock, you want to know something? Everybody's human.
Spock: I find that remark ... insulting.
Technically, yes (and it's really something he should have known, since Spock had similar moments in some of the TV series episodes), but especially given his closeness with Kirk, Spock would probably have just put it down to him being a silly emotional human, and not taken it personally. (Also, by the time the movies come around, Spock has started to embrace his human side, so it's unclear how much of this objection is true offense and how much is just banter.)
And of course, when Spock says, "I find that remark... insulting," he is likely just making a joke at Kirk's expense.
The Ceti Eel that crawled out of Chekov's was much bigger than when it went in. Yuck.
Khan did say that those buggers grow inside their victims' heads.
And this growth is what makes the victim susceptible to suggestion. Khan's mind control relies on dealing brain damage.
Khan was able to cause so much pain and trouble for Kirk, largely because when he was younger, he sailed off into the sunset without giving any thought to following up on Khan and his followers. As a result, Kirk is blindsided by a foe from fifteen years ago. How many other enemies bearing grudges could Kirk have forgotten about over the years, just waiting for the opportunity to strike back at him and his?
The idea of unknown threats lurking in the darkness ready to strike out at the unsuspecting Federation or the heroes for perceived wrongs has come up more than a few times across the franchise, to include Beyond and Lower Decks.
Carol and David Marcus fully expect Starfleet to try and and steal project Genesis. Think about what that says about Starfleet's sense of ethics that people expect them to steal a planet destroying superweapon.
You know that triumphant scene when Chekov returns to the bridge of the Enterprise and mans the weapons station for the final phase of the fight against Khan? Since Khan's on the Reliant, he's currently firing on what was very recently his own ship.
It gets better/worse. According to an above entry, some of the crew was still on there under the sway of the Ceti Alpha Eels. Then again, given the way the eels worked, death would've been a Mercy Kill.
Except, also as above, there were references in the movie that the Reliant crew were on Ceti Alpha V.
The Genesis Device and project serve as a relatively rare two-fer. When it comes to present-day nuclear weapons, the drawbacks to their use include the fact that they render the territory they strike uninhabitable for an indefinite period of time (so they cannot be used for territorial gain), the potential for the fallout to eventually harm the territory and population of the aggressor and the threat of mutually-assured destruction. Using Genesis as a weapon would exterminate all existing life on the target, but replace it with an unspoiled, life-supporting "blank slate" ripe for colonisation. Interplanetary (or inter-solar system) conflict means that its use would be unlikely to harm the aggressor's territory in any way, and the technology is so new that no other species, race or alliance has it (rendering mutually-assured destruction moot). The writers have turned what might have been a regular MacGuffin into the highest-octane Nightmare Fuel when applied to present-day thinking, and when David Marcus says their creation could be "perverted into a dreadful weapon" he is arguably significantly understating the case. Then we have the fact that Khan Noonien Singh is a ruthless despot who firmly believes in the racial superiority of his people, and unlike the most infamous example of that in recent history Khan is very skilled in military tactics and strategy. Not only that, but he almost certainly has the intelligence to reverse-engineer Genesis technology were he to get his hands on it. The potential consequences for every other living being in the galaxy would be horrific beyond description.