If I start overtime with an on-side kick and recover it, can I then win with a field goal? Does the other teams flubbing the on-side kick count as their possession?
Yes, a kicking play gives the receiving team an opportunity for possession which they have flubbed. It’s an unlikely strategy because failure to recover the onside kick by the kicking team leaves the receiving team with excellent field position. The same reason onside kicks are rare in regular play.
Game’s over. From the overtime rules in the NFL rulebook:
So an onside kick recovered by the kicking team means that the receiving team has had their required opportunity. Also, it appears that if you’re the first possessing team and kick a field goal, then do an onside kick and recover it, the game is over as well.
Another scenario - you get the ball first, don’t do much with it, and punt on 4th down. The other team muffs the punt, and you recover. At that point all you need is a field goal to win - they’ve had their opportunity to possess the ball, so you don’t have to kickoff to them again after a field goal.
Another scenario. You receive the opening kick-off. Drive down and get a field goal. Kick-off to them. They drive down to your 10. Fumble the ball, you pick it up run it back 10 yards, fumble it, they pick it up and run it in for a touchdown. Do they win, or was it over when you got possession? And what if they don’t run it in. They just regain possession on the same play? Is it over or do you keep playing?
Here’s the notes from the NFL rulebook:
It’s a little confusing, but the way I’m reading says that basically, with the double fumble, the game is over unless they score that play.
The way I’m reading it enalzi, is they lose even if they regain possession and score on the same play, after the second fumble - that’s what they mean by “the down will be permitted to run to its conclusion, but if the second team’s possession has legally ended with the fumble recovery or interception, any subsequent action will not affect the outcome of the game.” and “or a subsequent loss of possession by the team that intercepted the pass or recovered the ball, cannot change the result” there’s no reason for those clause except to cover the case of what happens after the first change of possession during the same play - if the play ends with the ball in the possession of the first team, of course the game is over, no need for a note to cover that.
You’re right, I misread “second team’s endzone” to mean the endzone they need to get to.
Another weird part of the overtime rules is that a safety will end the game instantly. It seems odd that a 3 point field goal will give the opponent a chance to respond, but a a 2 point safety will instantly win you the game just as a touchdown would. But that’s how it works.
As to doing an onside kick, I could see a team trying that on a kickoff if they know they have a weak defense and fear giving up a touchdown on the first possession, but a strong special teams unit and offense that both has a good chance to recover the kick and consequently score. That would have to be a really unbalanced team though.
That’s no more odd than a touchdown ending the game. The only reason for the current Field Goal rule was to stop OT from being a coin toss. It was far too easy for the receiving team to score a field goal and win on the first possession. Otherwise it’s still Sudden Death, you score you win. The rule has only made it a slightly more fair duel instead of allowing one of the parties to fire the first shot from arm’s length. What is an atrocity is having a new coin toss before OT. If the coin toss at the beginning of the game doesn’t determine who controls the ball in OT than it’s just a pointless gesture, might as well just determine who starts the game based on being home or away, or who won baseball’s All Star game.
You have a good point. Although only worth 2 points a safety is a heck of a lot more difficult and rarer than a field goal. It just seems ironic that a lower score will win when a higher score won’t, that’s all. 
Yeah, the opening coin toss means very little. And some teams always defer while others always want to receive first so at times it literally means nothing, it becomes “heads I win, but so do you”.
I think the NFL rules and examples were written by IRS agents who thought the tax code wasn’t quite confusing enough.
Getting back to Enalzi’s hypotheticals, let me ask this. Team A gets the ball first in OT and kicks field goal. A has had its chance to score, so that if B doesn’t score on its possession, A wins. B throws interception to A, and A’s player tries to return interception and fumbles back to B. Is the game over when A intercepts, in which case the ref should blow the play dead, or does it continue through to B’s fumble back to B, and B gets ball back with another chance?
I say under the “Play through the whistle” theory, that the ref should only blow the play dead when the play is over normally, so that B would get the ball back. Am I right?
In OT a safety does seem to lose much of it’s meaning. In regular play a safety not only scores 2 points, but the ball is returned to the scoring team through a kick. It’s a very valuable score that way, essentially stealing a possession from the other team on top of the 2 points scored, but that in itself makes it more valuable in OT even if it were treated like a field goal because it still eliminates the other team’s chance to score. In a shoot-out style of OT it would be the ideal winning play.
There’s two things to that scenario. First, A needs to end the play with possession of the ball, so the play isn’t blown dead at the interception, because B could recover before the end of the play as you suggested. Fumbled interceptions are definitely not unheard of. Also, turnovers are routinely reviewed and no more so than if the turnover can decide the fate of the game, so the interception doesn’t instantly end the game anyway. Player A would be wise running out of bounds immediately if he intercepted to end the play and avoid the chance of fumble (which is made more likely by contact).
But if A ends that play with possession and the turnover is confirmed then the game would end at that point.
EDIT: Replying to zamboniracer.
Well, the original rule was extremely straightforward: “Sudden death, the first score ends the game immediately.”
The revised rule, in theory, is fairly simple, too, since it essentially boils down to, “if the team which receives the initial OT kickoff scores a field goal on its first possession, the other team gets a chance at a possession.” It’s all of the corner-cases which that fairly straightforward idea brings to mind which cause the tax code rules. 
That listed in the same notes enalzi and I were referring to, and it doesn’t matter if the change of possession happens due to an interception or fumble - even though they let the play go to the end, nothing that happens after the change of possession matters. B had possession, their possession ended, A wins.
A scores FG first. Kicks off to B. B throws an interception to A, which muldoonthief says means A wins the game as it scored first in overtime, B had its chance but turned it over to A. What if B throws an interception to A, A fumbles the interception return, and B recovers and runs it for a TD? B would win, I think. Agree?
Nope. Team A effectively won as soon as they gained possession of the ball with the interception. The play needs to run to conclusion because there isn’t a rule to whistle a play dead simply because of that situation, and determining whether possession actually changes isn’t an instant decision in the NFL anymore, just like any other reviewable call. But the rule explicitly states that nothing following the change of possession will affect the outcome of the game.
I see how you arrive at that decision TriPolar, however, I’m thinking the NFL is asking for a riot if Team B is the home team, B scores on the last play of OT and A nevertheless wins. The NFL’s interpretation, if Tri is correct, is one of its more bone-headed.