A ball that lands inbounds (fair ball) and then bounces over a fence or other barrier is ruled a 'ground rule double'. The batter is required to stop at second base. All other runners are allowed to advance two bases or to home plate, scoring a run.
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If the ball hits the ground it cannot be called a strike. It is a "ball". It can still be hit like any other "ball". It would be like hitting a high, low, inside or outside pitch. Any "ball" can still be put in play.
I am under the assumption that the fielder must come down with the ball in fair territory for this to be an out. If he catches the ball while he is jumping over the fence and lands out of the playing area then it is a home run.
Correct or Incorrect?
If the ball hits the top of the yellow fence it is a homerun, even if it comes back onto the field.
It would be an out. It is still in play and is considered a flyball which if caught in the air, the batter is out.