Other mistake: When the saucer of the Enterprise finally crashes, all sorts of dust comes into the bridge through the big window at the front. Two mistakes here: 1. The very next shot shows that the dust cloud never actually reaches up to the bridge (which is in the middle of the saucer) and 2. Later in the movie, Kirk and Chekov escape the bridge, but to do so, they need to shoot the window out. So the dust couldn't have got in even if it had reached the bridge. Which it didn't. (00:32:25 - 00:51:30)
Other mistake: When Krall boards the Enterprise, he lands his ship in the top of the saucer, but then later, Uhura separates the saucer from the neck of the ship with both her and Krall still inside it. And then when they are back on the planet's surface, they arrive there in Krall's very recognizable three-nacelled ship, whereas all of the other ships from that fleet only have two nacelles.