Does the run score

[youngump]

Originally Posted by Dakota

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.

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?