Battles Won X Times Out

No.

Yes, it could be.

I'm not sure if a pre is that the defender should have defeated at least one pokemon or not.

How I know? In the begin release of Blissey I failed miserably against some high CP one and one of my Pokemon got beaten and subsequently I timed out. Some time later I noticed it had a battle won below it's name.

Yes, I'm very ashamed.

Yes, it does. The count is just inaccurate and glitchy. Even when fainting the last pokemon it does not go up everytime.

My chansey got a battle won when it was the only one in the gym pre gen 2. There's absolutely no way it cleared a whole squad.