I would go with the wood. It will paint & caulk better. Some of the PVC stuff doesn't cut well on a chop box, has a tendency to break and may crack with larger nails. PVC just looks cheap when its up, you can tell its plastic.
If the composite is MDF, I would buy that over wood. MDF is cheaper, and usualy straighter pieces. It also paints nicely. The only time I might choose wood over MDF is if I were staining the molding.
The paper is a key part of the drywall structure. Just as when you cut the paper on the drywall it's easy to snap, when you screw past the paper it's easy to blow out the back. Plaster and paper are a lot like concrete and rebar, the plaster based core of drywall resists compression, while the paper resists tension.
It's not essential to pull the old screws, the damage has already been done and will be mudded over. But you should add a second screw a few inches away anywhere you went too deep. This is a critical repair for drywall on the ceiling, and a very good idea for drywall mounted on the wall. I wouldn't go through any added effort to tear down the drywall, since it's still perfectly good. Just add the extra screws and be happy you caught the problem before experiencing a collapse.
Update: From your new photos, those screws are too deep. Once the paper has been torn, you lose strength at that screw location. You should never be able to see plaster around the screw head, but you also should never be able to run a flat edge over the drywall and feel the screw head above the drywall. It's a fine line to walk, but a professional drywaller should be able to walk it with ease (we train amateurs to do this in under a hour).
Also, from those photos, it doesn't appear that the installers are using a drywall bit (we refer to them as mushroom bits because of their shape). They leave a distinct ring around each screw, making it easy to countersink the proper depth, and preventing you from going deeper (the bit will cam out when it hits the paper).
Best Answer
It sounds like you are trying to build something in one place, pack it up and re-build it in another. If this is the case then screws are the way to go for dry fit, and then glue and screws for the final instal. I generally use Course thread dry-wall screws - because I feel they bite and pull MDF the best.
The only thing here is that there is a limited number of times that you can use the same holes with MDF, the Fiber-board looses a little meterial every time you screw into and out of it. So if you are building in a shop and then transporting onto site and reassembling once you are fine, if you are going to build and then rebuild this many times it will get less stable each time you do it.
As a side note, I also like to use pocket-hole jointery a lot because I like that it hides the screws from view.
Also, whenever I build boxes to be re-built I always make sure that I have some small square blocks with me. I use these blocks to make sure the inside of the box is square, just screw them in and it helps to keep the box from racking side to side.