Factual error: At one point Khan threatens to target the Enterprise's life support systems which are located behind her "aft nacelle". The Enterprise doesn't have an aft nacelle; as the many exterior shots show, it has a port nacelle and a starboard nacelle. Khan a) is familiar enough with Starfleet ships to know this basic fact about the Enterprise, and b) can see its nacelles for himself on his own viewscreen while he's delivering this line. There is no "nacelle" housing the impulse engines. They are enclosed by the hull of the saucer section.
Factual error: Enterprise and Vengeance come out of warp near the moon, 237,000km from Earth. Due to their altercation they lose power and proceed to fall freely toward Earth. The scene plays as though it takes a matter of a few minutes, too fast to get the situation under control. In reality the freefall time from that height to sea level is about 13 days and 6 hours; plenty of time to effect repairs or call port authorities for aid.
Factual error: During the Vengeance's fall to Earth, just after hitting Alcatraz, the saucer section hits the surface of San Francisco Bay, the outer rim diving dozens of meters into the water and its rear lifting up almost vertically into the air. A few seconds later the ship is seen from the onshore view, and it is straight and level and heading directly at Star Fleet Headquarters. Hitting the water at that angle and speed, so close to the shore, the ship could not have righted itself and gotten on that trajectory going that fast. It should have sunk or flipped over when it hit the water. Even if it could flip 360°, there was not enough time, distance, or height for the ship to continue on to its target at that speed. The water's drag would have slowed it too much.