New Official (Almost) with a Question

[FeetBallRef]

Folks, A few more questions & hopefully I'll be a happy camper.

If the Fielder leaves their feet in fair territory and catches the ball over the fence and then lands over the fence in dead-ball territory, it is still a catch correct???

If the Fielder reaches over the fence to catch the ball, the ball hits the heel of the glove and the ball then comes back over the fence in fair territory and hits the ground, it is a fair ball that is in play like any other ball hit into the outfield, correct???