Does this actually work for any proportions of A and B? Is there a simple proof?

Yes, but I'm not sure it's worth proving? I'd say that the "Factoring" section explains how this works, though there are no proofs. Will add pointers at the beginning.

Could we have arranged the rectangle so that B and -B were divided by a straight line w/o changing any of the probabilities?

This is addressed in the factoring section.