catch and carry

[Marty Rogers]

Rule 7.04: Runner advances one base when:
(c) a fielder, after catching a fly ball,
falls into a bench, or stand, or falls
across ropes into a crowd...

Where ever he leaped from (or touched last)
is his status of being in live ball territory
(LBT) or dead ball (DBT).
It sounds like the fence in play one is on the
line, so he came from LBT, was in the air,
made the catch, then tumbled out. This is a
catch even if he touched the fence on the way over.
Since you said he "tumbled," I envisioned him
falling. Ball is dead. If he had NOT fallen down,
the ball remaims live.

I would rule play two exactly the same.
If the fielder fell, advance runner one base.
If the fielder remained on his feet, ball
is live.