So that leads me to think, does the wood and other non magnetic factors of the build of the guitar affect the tone, the sustain, or both?
Sure - density of the wood, amount of wood, tailpiece, bridge, nut, even the tuning pegs.
And if I had identical strats, with everything adjusted the same except one made out basswood and one made out of swamp ash, would the average listener be able to hear a difference?
Are they really
identical? You could have two made out of the same wood and they'll sound different because it's a different piece of wood.
What's an average listener? Some might be able to hear a difference. An average listener wouldn't care.