Runner on 2nd no outs.Fly ball to center that is caught. Runner takes off with the pitch never tagging up rounds third. Center fielder attempts to throw ball to third but instead airmails it over the fence out of play. What happens to the runner?
is he out? safe at third or home. Can there be an appeal?

Dead ball.....R2 is awarded 2 bases from the time of the throw. So award runner home...however, runner must retouch 2nd or be liable to be put out on appeal...

Also, possible problem for the runner if he was still on or beyond third base when the ball became dead. If so, he can't legally return to second base to tag up and could be called out on that appeal.

Runner was around third when ball went out of play. How should the appeal be started?

Once the base running action has completed...under a dead ball appeal, a coach or any player with or without the ball can appeal the runner leaving early..

so the runner has no options if he passed 3rd already? sounds like an adavantage to the defense, even though both sides screwed up..

no, once he went passed 3rd, he could not go back and touch.....unfortunatly like many things in baseball not every situation is equal...or fair...he just has to hope that the umpire did not see him leave early....

I disagree, the runner was at or beyond 3rd when the ball "went out of play" he can indeed return and retouch 2nd as long as he doesnt touch HP. The ruling on touching the NEXT BASE is after DB. In this situation had the runner continued after the DB and touched home, he then could not retouch second.
A clearer example would be a fly ball to the gap that rolls under for a 2B award. The runner misses 1st base, reaches 2nd and beyond when the ball goes out of play (GRDBL), from that point the ruling starts, if he advances and touches 3rd on the DB, he can not return and touch 1st, if he returns from between 2nd and 3rd he is legal.
7.10b2, (NFHS 8-4-2q) (NCAA 8-6a)
From J/R:
A runner is vulnerable to a retouch appeal if
1. re has not retouched (easy one)
2. he has failed to retouch, the ball becomes dead AND "he then" proceeds to touch or pass an advance base.

EG: R1, one out. R1 thinking there are two outs, continues running "past 2nd base" as the batters fly ball goes towards the right fielder. The right fielder catches the ball for the second out and throws toward 1st base for an appeal of R1's failure to retouch. R1 is standing between 2nd and 3rd when the right fielders throw goes out of play: if R1 DOES NOT proceed to touch 3rd base AFTER the ball has become dead, he CAN return to touch 1st base, second base and 3rd.

This makes sense, a runner should not be penalized for his speed or anticipation of a ball dropping in, only for missing a base or leaving early.

jjk,
OBR and NFHS treat this situation very differently. In FED, it is just as piaa_ump has said-- see 8-4-2q and 8.2.5, or Casebook 8.2.5a. OBR, on the other hand treats this as you have described it. I'm not clear how NCAA handles deadball retouches. Carl Childress, in his Baseball Rule Differences, says " Let's be clear on something: the FED 'return after catch' rule
is much different from the OBR rule."

By the way, when J/R lists an OBR rule in brackets, that is the rule citation which justifies what they have written. Oddly enough, when a NFHS or NCAA rule is shown in brackets, it means that FED or NCAA handles the situation differently from OBR.

Three Finger and PIAA have it right. HS is different in this regard. It's a strange way to rule but it is the FED way.

Just another example of the joys we have while working multiple codes.....

On the R1 at first, he leaves early touches 2nd and is now between 2nd and 3rd. The fly ball is caught, R1 returns to 1st with out re-touching 2nd but get back to 1st before the ball. Is the appeal now to 2nd for failing to re-touch on the return trip? My belief is yes.

Yes, the defense can appeal that the runner missed second base on his return to tag up at first. The rules indicate that a runner must touch all intervening bases whether advancing or returning.

Yes, he subject to an appeal. I'm sure you have the axiom of "last time by." This goes for touches or no touches. If he touches on the way by but misses on the return , then he is liable. On the other hand, if he misses the first time but touches on the return then he isn't.

the reason the baserunner can not go back and retouch second if he has already touched third and the ball in dead is he would be guilty of running the bases in reverse order which he is not allowed to do.