Is ball "dead" after 3 outs... Situation

[BJ Moose]

Is ball "dead" after 3rd out of a half inning?
Situation:
R1... 2 outs. Batter hits GAPPER. R1 rounds 3rd, heads home, relay throw to catcher, a SLIDE, a SWIPE TAG attempt, cloud of dust... NO SIGNAL. (Assume: missed tag, missed plate). R1 gets up, moves towards dugout, F2 gets up with ball, sees ump and R1 and takes a step toward R1. Just now F2 notices BATTER-RUNNER take wide turn at third. Catcher throws to third and tag is made on B-R diving into third for THIRD OUT!

a)... Out call at 3rd comes 1 second AFTER R1 touches home. (This has gotta be easy, count this run now!)

b)... Out call at 3rd comes JUST PRIOR to R1 returning and touching home.
Do WE allow this retouch... or is the BALL dead? Do we allow a "4th out appeal?" Does catcher have to go get the ball, and make the appeal?