1. It will still move with the webcam, it will just not pivot - it will stay glued to the neck and (generally) make the neck bend to move the head around. If there is a lot of non-independent artwork, the warping might look funny (e.g. long hair or a hoodie). I think it works better with a narrow(ish) neck with rear hair independent - that way the neck bends (especially with sticks across shoulders and along jaw lines). But its a matter of trying it and see.
2. Yes, you can put handles on the one layer and tag them with different body parts. That is normal. E.g. knee, ankle, foot, toe is normally on the one leg layer for the walk behavior.
3. I worry that the hair will not stay synchronized with the head position if you move the head from side to side. But try it! If it looks good, it is good!
4. There are more advanced approaches, like what you suggest. E.g. you can clone the whole puppet tree (have two root characters) and put the rear hair in one of the two copies. Think of it like two people standing in front of each other - you can only see the one in front. But its hard to keep them in sync - e.g. handles need to be lined up perfectly. But webcam movements will go to both puppets, so it might work.