Any score that is not a field goal ends the game immediately -- TD, safety, or points awarded for palpably unfair act. You don't need to get into possession or the details of the rule if any of that occurs.
Otherwise, I think the NFL has answered pretty much all the scenarios one can think of in their recent guidance on the issue. Here's the link:
NFL.com news: Postseason overtime rules. Basically, the trickiest plays are kicks and muffs. A muff does not constitute possession, but it does constitute an opportuntity for possession if it's the receiving team. I think the blocked punt on first possession scenarios and explanations go a long way to toward understanding the entire rule:
A.R. 16.11 BLOCKED PUNT
Fourth-and-5 on A45. On the opening possession of overtime, Team A's punt is blocked by B3 at the A35. The ball hits the ground beyond the line at the B48 and bounces back to the A43 where it is recovered by A4 who runs to the B40 for a first down.
Ruling: A's ball, first-and-10 on B40. Team B is not considered to have had an opportunity to possess the ball because it did not touch the ball beyond the line of scrimmage.
A.R. 16.12 BLOCKED PUNT
Fourth-and-5 on A45. On the opening possession of overtime, Team A's punt is blocked by B3 at the A35. The ball bounces beyond the line to the A48, where B4 muffs the loose ball to the A45 where it is recovered by A4.
Ruling: A's ball, first-and-10 on A45. By touching the punt beyond the line of scrimmage, a new series is awarded to Team A. Since Team B had the opportunity to possess the ball, both teams have met the minimum requirements for possession.
In my view, they've created a problem worse than they had in the first place. The rule was, as I understand it, based on a perception that the team that gets the ball wins too often. If think if you could have lots of experience with the new rule, it would be revealed that the kicking team will win more.