Game-Winning Field Goals and the Clock

Factual Questions
1

When does the clock stop after a field goal is kicked?

An awful lot of them get kicked with 5 or 6 seconds left on the clock and I’ve never seen one that took so long that there was time for the opposing team to make a play.

Tonight I watched Al Del Greco’s game-winner vs. Philadelphia(?) in slow motion and the ball passed through the goal posts with at least two seconds left and then sailed out of the end zone with time still on the clock. Those seem like a couple of logical timepoints for the clock to stop but it just kept on ticking. Is that correct, or do the timekeepers just let it run out when it’s so near the end of the game?

2

I believe the clock runs until the referees indicate to the timekeeper that it should stop (up until the AFL-NFL merger, the NFL used to have official time kept by an official on the field; the stadium clock was unofficial).

3

The officials will stop the clock once the feel that the ball has gone through the uprights and they’ve made their call.
The referee usually decides that and he isn’t standing right under the goal posts.