Other mistake: On board the USS Reliant as it orbits Ceti Alpha 6 the systems panels show ship diagnostic diagrams including a wire model of a Constitution class hull, something the Reliant doesn't have. (00:13:01)
Other mistake: When Captain Terrell shoots himself with the phaser, the phaser disintegrates along with his body, when it should have just dropped to the ground (as often shown in the various Trek TV shows). It wouldn't make sense to explain that the phaser disintegrated because Terrell was touching it, because then it would be impossible to explain why nothing happens to the ground that people are standing on when they get shot (Director's Cut). (01:11:20)
Other mistake: As the turbo lift doors open for Joachim to say his line, the diagram of the lift system on the back wall is of the Enterprise's system.
Other mistake: On the Reliant, as the woman turns and goes to activate the transporter, the blue diagram on the monitor above and to the right of her head is the port and top diagnostic views of the Enterprise, rather than the Reliant.
Other mistake: After Khan activates Genesis, David says the device will explode in 4 minutes. However, on the device itself we see the countdown begin at 999 at a rate of 2 per second. This correlates to about 8 minutes, 20 seconds.
Other mistake: After the Enterprise blows the port nacelle off the Reliant, the destroyed Reliant bridge is shown. The three sheaves of wires hanging from the ceiling should be on a tilt with the angle of the ship rather than straight down. Even accounting for the artificial gravity unit, which could have been damaged.
Chosen answer: I imagine there was a memorial service for everyone killed. Starfleet's policy on corpses is probably to return them to Starfleet HQ where their families can collect them for whatever services or ceremonies they want unless the crewman had left instructions specifying otherwise. There's no telling why Kirk sent Spock's body to Genesis. Based on Sarek's reactions in ST3 he almost certainly went against Spock's wishes, unless of course, Spock left no recorded instructions and Kirk did what he thought would please Spock based on his being highest ranking officer and Spock's closest friend. It also seems very out of character for Spock to just assume that whoever he transferred his katra to would be able to handle it and carry out his wishes (McCoy certainly couldn't!). Ultimately it seems we have to chalk it up to a plot device to base the sequel on.
Grumpy Scot
According to the novelization, Kirk's intentions were to send Spock's remains into the Genesis sun. Lieutenant Saavik altered the trajectory of the torpedo beforehand, due to Spock's desire to see the Genesis effect for himself. The torpedo casing was expected to incinerate when entering the atmosphere. As pointed out by David Marcus in STIII when the pod was detected on the scanners, the gravitational fields were in flux at the time, and the pod had obviously soft-landed on the surface.