Does the head stay down and still? NO. If you look at the great players in the game, most have a slight movement of their head going back. It certainly varies from player to player, but it's still present in one form or another. Players like Jack Nicklaus and Tiger Woods have quite a noticeable head movement on the back swing (especially with their drivers). Other players, like Steve Elkington and Tom Purtzer, hold theirs a bit more stationary.
What's important to note with head movement, just as in the weight shift, is that it's not something you should think of. It is simply just a reaction to what you do with the golf club. When a golfer starts thinking of shifting their head a little bit here, and a little bit there, there is big time trouble awaiting.