We'll have to agree to disagree on that one, the strings have no choice but to vibrate with the piece of wood they're attached to, if the wood is capable of affecting the guitar's sound then the strings are going to be affected by the wood vibrating as well, they can't vibrate independently from each other.
Playing plugged in is really just testing the pickups, sure swapping pickups may save an otherwise poor sounding guitar (but not always), but a guitar that sounds nice unplugged can always be made to sound good plugged in
Agree, only if the body were perfectly rigid (impossible) would it not be modulated by the string vibration. In technical terms (sorry) the oscillations are governed by the wave equation which is highly dependent on boundary conditions. It's a coupled system and the material properties matter.