Does the run score
 

I have 2 interesting plays form this week. One happened to me and the other was told to me by a coach on a play that involved his daughter. Here I’ll give you the play told to me. It’s long.

Runners on 2B & 3B with 1 Out. Looping fly to deep SS and SS somehow gets back between hole and 2B and snags the ball. 2 Outs. Runners were not tagging. Runner from 2B well toward 3rd before turning and running back to 2B. SS tags her before the bag. 3 outs.

Meanwhile the runner from 3B went all the way home, touched the plate before the tag at 2B and was standing off to the side before she realizes the ball was caught. She starts running back towards 3B and the SS starts running at her when the umpire yells “Dead Ball. We have 3 outs.” This stopped all play and offense and defense left the field. Then the ump says “Run does not score.”

1st: I think ump should have not stopped the play.
2nd: I think this was an appropriate situation for a 4th out appeal on the runner from 3B leaving early.
3rd: I think the run should count unless there was a proper appeal for leaving 3B too soon. The 3rd Out at 2B was a timing play and occurred after the runner crossed the plate. (Unfortunately the umpire ruined the chance for what might have been a live 4th out appeal.

Even if the runner had beat the fielder back to 3B, so what. Doesn’t the run still score? Don’t we need an appeal to un-score that run? I think so because after the 3rd out, the runner can’t go back to 3rd to tag up and then re-score. Maybe a simpler question is after the 3rd out, is there anything the offense could have done to negate a 4th out appeal and still score the run? (Sorry, messy situation.)

Sure... retouch home plate (doesn't sound like the runner did that before returning to 3B), retouch 3B, then go home again, before runner from 2B is tagged.

Additional information:
If R1 returned to touch 3B and does not retouch home on the way, there is no longer an appeal to be made at 3B. There is an appeal available for missing home on the way back to 3rd, however.

But after the 3rd out, can R1 return to 3rd? I think whatever she does has to be completely done before that 3rd out (as jmkupka says above). After the 3rd out even if the runner retags home, runs back to 3rd and then goes back and steps on home it does not matter. There were already 3 outs and if a 4th out appeal is made it should still cancel the run. If no 4th out appeal is made, no matter what R1 does, I think the run scores.

I'm not sure this is right. If the runner retreats past home to retouch, she hasn't scored anymore, right? If she hasn't scored then she can longer be called out on appeal (it's not an advantageous 4th out).

But to me the question remains, how long can the runner unscore.

Suppose for example that there's one out and it's a ball to a very very deep rf. The runner thinking there are two outs breaks on contact and makes it home. Then the RF catches the ball. She can unscore by running back to third (touching home along the way) and if she gets there safely, she's staying on third.
Now same scenario but there are two outs and after scoring the coach thinking there's one out calls her back. She touches home either a) just before) or b) just after the catch.
Rulings?

If there are 2 outs and the fly ball is caught in RF it doesn't matter what the 3B runner does. Even though she may have crossed the plate before the out, you can not score a run if the 3rd out is a force or a runner put out before reaching 1st.