6. Spock's Character Backtrack
So I think it was well established by the end of 'Star Trek' (2009) that Spock had earned Kirk's trust and visa-versa. So why did he file a differing report to Starfleet at the beginning of the movie that he knew would certainly get Kirk reprimanded, a breach of protocol that saved Spock's life I might add, without even consulting his Captain? Did something happen between the two of them between the events of the two movies that would cause some friction? He even had his future self tell him outright that risking many time paradoxes was worth it to make sure that the two would work together for a greater good, that the bond the two formed would be invaluable, how much more logical convincing does the guy need?
5. Khan Off His Mark
His newly commandeered vessel crippled, without weapons, thinking that his all his comrades are blasted away in torpedo explosions, Khan goes for his final stand... against StarFleet HQ? He is seemingly able to pilot the ship enough to aim it right at San Francisco, so how about the Enterprise? You know the ship that is actually responsible for Khan's most recent misfortunes, including seemingly murdering all his crew? In fact, he almost hits the Enterprise going in for the kill against Starfleet HQ killing thousands in the impact. He couldn't aim just a little more to the right to take out the one ship that has done him so much wrong?