LL Play- Appeal Question

[lpneck]

R2 didn't tag- he went part way to 3rd on a medium-depth fly ball to CF. He started going back to 2nd base once the ball was in the CF's glove, but then when he dropped the ball on the transfer, he took off for 3rd and never went all the way back to the base.

In the end, they didn't rule it a catch, so it didn't matter and R2 scored.

Does the ball being live or dead matter on the run scoring? Let's just simplify the play for a minute without the confusion of no one knowing whether the umpires had ruled a catch.

Runners on 2nd and 3rd and 1 out. Flyball to CF that is caught for out #2. R3 tags up and crosses home plate. R2 does not tag up and advances to R3.

a.) The CF throws the ball to the SS who steps on 2B to appeal the play.
b.) The ball goes back to the pitcher. Time is never called. The pitcher steps off the rubber and throws the ball to the SS who steps on 2B.
c.) The ball goes back to the pitcher. The defensive coach asks for time and goes out to the mound to explain to the pitcher they are going to appeal and he needs to throw the ball to 2nd base. After the coach leaves the field, the umpire puts the ball in play, and the pitcher steps off the rubber and throws to the SS who steps on 2B.

Any differences in those plays? Or would the appeal always would have to occur before R3 crosses home plate in order to prevent that run from scoring?

BTW, this was the state semifinal game the day before the crazy scenario in the other play in the championship game.