If the ball hits home plate then hits batter are you out?

Answer 1

If the ground ball is foul, you are not out.

If the ground ball is fair, you are out unless you are touching your base (however, if you are touching your base and in the umpire's judgment you intentionally interfered, you are out).

Rule 7.08(f) states:

Any runner is out when "he is touched by a fair ball in fair territory before the ball has touched or passed an infielder. The ball is dead and no runner may score, nor runners advance, except runners forced to advance. EXCEPTION: If a runner is touching his base when touched by an Infield Fly, he is not out, although the batter is out."

Answer 2

No, on a throw the runner can't just be hit by the ball and be out. Even if it hits the plate. The catcher would have to tag the runner with the ball, or with his glove while the ball is in his glove. The only time a runner can be out like that is when they are running the bases and hit by a ball hit by the batter(The runner must be in fair territory).

Wrong. If you are a right handed batter, it is possible to hit the plate then hit you.

You are out. If you are out of the box.

If you are in the box, the ball will called dead. Scored as a foul.