A friend wouldn't have filed that report. Or would have at least told Kirk beforehand. Spock had to have known what that meeting with Pike was going to be about. But nope, no warning at all! Not even some speculation about the possible ass reaming Kirk was gleefully walking into.
For how different the characters are, I think you need more time for the friendship to mesh naturally. I think AFTER the sacrifice, you can start to see it. And then you would hopefully, in the next movie, see some different reaction to each other. But no one, after what Spock did, would specifically ask for him back as first officer. How could you trust someone, regardless if he was actually right in what he did?
As for Uhura and Spock, I don't think a pity fuck is enough to base a relationship on. I mean the characters, why should they be together? When Uhura isn't playing comm officer, she's just nagging Spock about stuff, during critical missions(another dumb script piece). Do they both like skiing in Tahoe? Collecting stamps? Analingus? Cmon, give me something, otherwise they're closer to a couple that's just itching for a excuse to break up.