The pieces on your door that you're referring to are called map pocket cover, storage pocket lid, door pocket thingys, etc. Arm rest flaps work just fine.
Your car has what's called a "Soft Touch Interior". The plastic trim pieces are coated with a thin layer of a soft substance that scratches, flakes and mars easily, leading to a dogged out interior. Porsche did this to the Boxster starting in MY 2000. Other manufactures have also used the same goop.
Unfortunately there's really no way to protect them or clean them up once they've been dinged up. Myself and several of the other guys her on the board have replaced them with earlier '97-'99 pieces that lack the "protective" coating. I've also experimented with stripping the coating with various success. I've read stories of guys successfully sticking them in the dish washer and removing the coating that way, softening the coating up w/ a heat gun and stripping it off, etc.
If you commit to stripping the coating, be careful that you completely remove all of it and prep the surface before painting, as any coating that you paint over will eventually scratch or peel off.
For me, after dealing with the situation on several cars, it's least painful for me to just purchase the earlier non-covered pieces new from Porsche or trying to find good used pieces off eBay.
Good luck!