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.
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 Seti 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.
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.
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, 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.