I picked guilty gear for her since it's the most relevant series in my head currently, but I'll admit I've imagined her a bit in any fg I spend a lot of time playing. When I've talked about this stuff with friends we typically start the conversation with "Ok, what if your OC was in "X" game?" and go from there. The dream come true would be an fg with a ton of my friends' and my own characters tossed in, think mugen but balanced.
As far as Vesh's reason for fighting? She simply does it for the thrill of combat and to further hone her craft. "Friendly competition with benefits" being her favorite, if you get my meaning.
Theme wise, I have two that come to mind:
