Warp, Seems Like Impulse
Checovs Fall
Switching Hands
When they go in to warp in the Klingon ship in the earths atmosphere, several minutes go by, then you see the ship again just outside the earths atmosphere. Ive seen space shuttles move faster than this.
When Chekov falls off the Dock, if you watch where he falls you can see waves moving away from where the stunt person fell in the water, but then the scene changes to show him lying on the dock.
In the scene where Kirk goes to the hospital for Checkov, he takes all the doctors and puts them in a back room, he then takes his phaser and proceeds to melt the lock to keep them in. When he walks towards the door the phaser is in his left hand and then in the close up shot it's in his right, when he's finished melting the lock and the camera is at a full body shot, the phaser is back in his original hand . (it's either like that or vice-versa, right hand to left).