I would definatly suggest getting some type of filter. I know you said before that you might get a rubbermaid type tub, are you still going to? That would help with the dirty water.
Im not exactly sure about the shell. Some peeling is normal, but you could get some turtle wax type of stuff to put on just to help, im not sure what it is called though. Do you have a calcium supplement for him?
As for the wood, I would take that out. Just use some bricks for now, you can clean those easily. I would put them under the basking lamp.
Found this link, seems pretty helpful.