If it turns, what turns it? It rocks back and forth with the lever it's attached to. Does it stay in mesh with a driving wheel during the rocking? It is rotated by a pinion hidden behind the rack. It moves away from the rack, thus releasing the rack, then moves back into place to count the strike.
Is the gathering wheel actually raising the rack? No, a pin behind the hour wheel raises the rack.
What holds the rack from falling after the last tooth is gathered? The wheel? Look at this video
. It is the same one at the very beginning of the post, but I have managed to rotate it to its proper perspective and slowed it down.