If you recover an onside kick, then the opponent never gained possession of the ball. So I imagine that if you failed to score again after recovering the ball then the other team would have a chance at a possession. If you did score again (another FG or a TD) before turning over the ball I expect you’d win at that point. I’m not sure how the NFL would handle two FGs before the other team gets the ball but I’m guessing they’d end the game.
It’s almost a moot point because it’d be insane to attempt it. Onside kicks are very rarely recovered so you’re almost gifting a short field to your opponent who can win on a TD. It would indicate having absolutely no faith in your defense.
Plus, if you were that desperate to keep the ball away from the offense, you wouldn’t have kicked a FG in the first place; you’d have done everything to try to get a TD including an attempt on 4th and long (which would be more likely to succeed than an onside kick). I can’t see any situation where this would ever make sense.
No, again the point of an onside kick is that you prevent the other team from possessing the ball. If you did an onside kick as your starting kickoff (which is legal; insanely risky, but legal) it would be no different than if you’d received the kickoff. After all, an onside kick is like a kickoff to yourself.