Home run or catch

It depends where the fielder was when he caught the ball. This is a "have to see it" call.

No different than when a fielder catches a foul ball and tumbles into the stands.  I believe the rule is if any part of the player is in play when he catches the ball, it is a catch and not a HR.

I have a catch.  If there are runners on base and the catch is not the third out, all runners are awarded one base.

From the NFHS definition of a catch:

"It is considered a catch if a fielder catches a fair or foul ball and then steps or falls into a bench, dugout, stand, bleacher or over any boundary or barrier, such as a fence, rope, chalk line, or a pre game determined imaginary boundary line from the field of play."

From the comment in the case book:

"It is a catch when he has one or both feet touching the playing field, or with both feet in flight prior to his touching any dead ball area."

Same answer in NCAA and PRO too