How would using a wooden bridge and the strings through the body would affect the tone and sustain of a bass? Would it depend on the type of wood used?

I have used wooden bridges, and will continue to on some of my fretless basses. They do give you more of that "upright tone", being less definition, and more of a "thud" sound. I wouldn't recommend them if you are wanting the "mwah" sound. The type of wood will effect the tone. I would recommend a very dense wood, since the wood will fluctuate and change your action.

I also use wood nuts on my fretless basses that match the fingerboard wood. It gives the open strings a closer sound to a "fretted" note.