Yes. MLB Rule 6.05(g) states that a batter is out when "His fair ball touches him before touching a fielder". If the fair ball touches the batter before it touches a fielder, the batter is called out.
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When a batter hits a ball and then leaves the batter's box, he immediately becomes a base runner. Rule 7.08(f) states that a runner is out if "He is touched by a fair ball in fair territory before the ball has touched or passed an infielder." I suppose a wildly spinning, batted ball could bounce towards the first base line and touch the batter-runner while he was running to first base. In that case, the batter - runner would be out.
Satchel Paige loved to tell the story of how "Cool Papa" Bell once hit a screaming line drive that hit Bell as he was sliding into second base. In that case Bell would clearly have been out.
If the ball hits the batter in fair territory, the batter is ruled out. If the ball hits the batter in foul territory or the batter's box, the call is a foul ball.
No the batter is not out- The bag is on the ground and that is considered a bounce off the ground.