Continuity mistake: When the Enterprise leaves spacedock, Saavik orders 'one quarter impulse power'. However, in the exterior shots the impulse engines are not lit. The reason is that ST II uses stock footage from ST I for the spacedock scene. In that movie, the ship leaves the dock using thrusters. The obvious give away is the blue deflector from ST II forward, while ST I had a red deflector. It is a short scene less than 2 seconds, you really have to watch for it. The thrusters are on, as seen in the parts of the scene with the blue deflector, and suddenly they are off and the deflector is red for a few seconds.
Continuity mistake: When Khan contacts the Enterprise after Reliant's opening attack, Kirk has his uniform jacket unbuttoned to about mid-chest level (a little above his stomach). When Khan demands all information on Project Genesis be turned over, and Kirk responds with 'Genesis? What's that?' his jacket is now unbuttoned all the way down to his belt.
Continuity mistake: When Captain Terrell kneels down and says "What the hell is that?" Chekhov's gloved hand can be seen on the left side, but in the next shot Chekhov is already in the next room.
Continuity mistake: When the Enterprise is making a run for the Mutara Nebula, they show a scene in a photon torpedo bay. On the wall the sign says "TORP BAY 4." The problem is that the Enterprise only has 2 photon torpedo bays. Bay 1 (Starboard side) and Bay 2 (Port side). This version of the Enterprise has no aft photon torpedo bays.
Continuity mistake: When Admiral Kirk and party arrive in the transport pod, they dock on the port side docking port in the secondary hull. But the airlock they emerge from is in the docking port for the port side torpedo bay, as signified by the sign in the background saying "TORP BAY 2." You can see the docking port they are supposed to have docked at to emerge from on the side of the torpedo launchers, just above of the secondary hull. The reason for this is because they re-used the stock footage from ST:TMP when Kirk and Scotty docked at the same docking port.
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.