Other mistake: After the Enterprise is disabled by the Vengeance and falls out of warp, Kirk and Spock have a discussion in one of the corridors about letting Khan help incapacitate the attacking ship. Watch behind them. The ship is running under Red, but the crew members are walking past them calmly, some even watching their conversation as they pass! Not really Red Alert protocol... In fact, through all of this, only one crewman runs past in the entire sequence.
Add timeMatt Yeary
Other mistake: As Kirk realigns the warp core to restore power and save the USS Enterprise from crashing to Earth, Sulu tells Spock: "Without power, we'll incinerate on re-entry." It shows Kirk restoring power after it plummets below the clouds, which is well beyond the point where they would have been incinerated.