Can you be in motion during a faceoff in hockey?

Answer 1

The ref usually waits until all players are in position before dropping the puck. No player can go offside (cross onto the opponent's side of the faceoff) or get too close to the faceoff dot before the puck is dropped. The center (the player taking the faceoff) cannot have a false start. If any of this happens, the center is kicked out of the circle. If the player replacing the center is kicked out as well, then it is a delay of game penalty.

Answer 2

No, though some officials will allow the slightest of motions.