Miss Base
 

R1 at 2nd, base hit. R1 touches 3rd and then the plate. The coach at 3rd yells that she missed 3rd. R1 touched the plate and is returning to 3rd when she is tagged out by the 3rd baseman. What do you have?
Same situation but she missed 3rd when the ball was hit. What do you have?
NFHS, NCAA and ASA rules.

A legally scored run cannot be "unscored." If she touched 3rd and home, then going back toward 3rd base does not make her liable to be put out.

Then, not legally scored, assuming still live ball, tag is an out.

My initial reaction was to agree with you and on the field I'll definitely be sticking with that. That said, for the purposes of discussion and understanding only:

R1 on 3rd is off with the pitch when B2 hits a monstrously high fly ball. Thinking adroitly, R1 continues home, touches home and then returns to 3rd where she is standing when the ball is caught.
She has touched all the bases without being out so why is this not a run under 5-5-A-1. (And this ignores Irish's regular point about the rules not even spelling out the order of the bases.)
For some reason, you've decided that a runner can unscore her run by retouching home and then 3rd if she missed third but not if she hit it. Is there anything in the rulebook to support that position? (With which I'll say again, I agree. It's clearly what they MEANT to write.)

Not a legal score if R1 left 3rd before the catch, just an attempt to advance.

Speaking ASA

Any runner who has touched all four bases has scored a legal run whether all bases are touched or not. Can someone please cite how an illegal run occurs?

A runner returning to touch a missed base or a base left too soon must touch all bases (home plate is a base)

In first scenario, I have no call to make saving the possibility of INT if it affects the defense not playing on another runner (8.7.P).

In the second, the still active runner can be retired by being tagged.

In the initial situation we have a run scored, then a potential for interference if the act by R1 of returning to third interfered with the defenses actions.

If the runner in question missed 3rd base we have an out when she was tagged. In this case her being tagged constitutes a live ball appeal of the missed base.

The question I have is what if she missed third, touched home, retouched home, and returned to third before being tagged. Can a runner who has crossed home plate retouch and return to a missed base (or left early if a fly ball). I want to say yes, but I don't remember.

If still in live ball area.