At first, it seems unfair Uhura is left behind. But it actually makes perfect sense: if you're hijacking the Federation's flagship and heading to an uninhabited planet, why would you need a communications officer? You'd have no want or need to talk to anybody else.
Also, after beaming to Enterprise, Kirk offers everyone else an out "Dr. McCoy and I have to do this, the rest of you do not." The crew stands by Kirk and goes with him to Genesis. They'd all just broken McCoy out of Starfleet's funny farm, so had to vamoose one way or another, Uhura hadn't done anything (except lock "Mr. Adventure" in the closet). Kirk hadn't planned on taking anyone else with him in the first place.
The novelization also adds that she helped to cover their escape with technological trickery, leading to her taking refuge in the Vulcan embassy with Sarek - legally, she would be considered an accessory regardless by helping them get to the Enterprise, knowing they intended to steal it. Might as well clear the board long enough for the others to get to the bridge and power up the ship without anyone questioning what's going on.
In the previous film, Spock sacrificed himself to save the Enterprise. In this film, Kirk shows he values his friends more even than his ship, and sacrifices the Enterprise to get his friend back. In the end, Kirk couldn't avoid losing something he cared deeply for, because one way or the other he had to suffer a humbling loss to lose the idea that there's no such thing as a no-win scenario.
The first two films had the Enterprise power up its engines and thrust forwards to clear its moorings. This time, the Enterprise instead flies backwards until it exits Spacedock, while the pursuing Excelsior exits Spacedock in the usual forward motion. While this realistically was just based on where each respective ship was parked inside the complex, thematically it represents that unlike the first two films, this flight of the Enterprise was not sanctioned by Starfleet.
Imagine what would have happened if none one realized that Bones had Spock's katra; Spock would die again when the Genesis explodes and Bones would be driven to insanity or worse.
Although we keep calling the boy that they rescued off the Genesis planet Spock, in actual fact he wasn't. He was an individual with completely different experiences, memories, thoughts and feelings to our Spock. We can argue all day as to who the rightful owner of the body is, but the fact remains that when our Spock gets that body back, they murdered the person that was already there without a second thought. And worse, one scene that was thankfully cut from Star Trek 4 has Saavik revealing that she actually became pregnant thanks to her helping him through the Pon Farr - a child that technically isn't our Spock's but the Spock who is now dead. Imagine waking up and discovering that you not only had sex with a work colleague without wanting to, but that you impregnated her. Considering we never see Saavik again (her role being cut from Star Trek 6 in favour of Valeris) it might still be true for all we know.
Some of that particular horror can be offset by the fact that this Spock (for lack of a better name) was, in many ways, a blank slate - the Genesis effect rejuvenated Spock's body, and then accelerated his growth so that, from the moment Saavik and David discover him, appearing about six or so, by the time that the bird of prey flies off and Genesis destroys itself, he's now about the same age as Spock was at his death, so bare minimum, a good forty years of progression, in the span of a few days at most. And, based on the connection that his body had to Genesis, the fact that he was unconscious from the escape to the delivery at Vulcan... It's entirely possible that the "mind" of the body recovered at Genesis died with the planet, leaving the body brain dead.
At some point, someone states that the child's mind is a "blank slate." This Troper took that to mean that Spock's mind was split between the child and his "Katra" in Bones, like a brain's two lobes being separated. It seemed like Spock's higher consciousness was in Bones, while the simplistic natural reactions were in the child's body, and that all the ceremony on Vulcan did was re-merge the two. So the child didn't necessarily "die."