Home Run or Catch?

[Baseball_North]

Under-18 game. Bases loaded. 2 outs. Batter hits a long fly ball to deep right field. The right fielder at the warning track jumps as high as he possibly can. He catches the ball in mid air while most of his body is still on the playing side of the fence. His momentum from the jump and stretch takes him over the fence. Somehow the player didn't hurt himself. He retains possession of the baseball but is now on the other side of the fence.

Is this a legal catch or should the batter be awarded a home run (grand slam in this case).

If this was a legal catch, what would be the situation be with less than 2 outs? Would the ball be considered dead? All runners advance 2 bases? All runners advance 1 base?

Thank you.