if you just extend the metaphor in the diagram, and imagine a negative length to just refer to direction, sure it does :)
personally, I love visual proofs because they can communicate an idea efficiently, sure they have their pitfalls, but its less about the actual mechanism of the proof and more about the core idea that lets me appreciate how its working- and visual proofs add a pseudo-physical intuition that helps me appreciate it.
Trying the proof with a < b, with the b square from the bottom-right as in the diagram, I get a region to the top and left, and moving a piece (differently to the diagram) I get (a + b)(b - a) as a positive area for that region, and then flip the sign because it's negative.
Not really. If b > a, then swap them to conclude that b^2 - a^2 = (b + a)(b - a), which is what the visual proof demonstrates.
Your conclusion is equivalent to saying that a^2 - b^2 = (a + b)(a - b).