It is a dead ball and the batter is awarded first base and is ruled a hit by pitch
If there are less than two outs, yes. If there are two outs, this is a timing play. If the runner crosses home plate before the batter is thrown out at second base, the run counts. If the batter is thrown out at second base before the runner crosses home plate, the run does not count.
Yes. If a runner crosses home plate before the third out is made (unless it's a force out), the run counts. For example, say the Yankees have runner on third with two outs. The batter hits a ball to the gap in right center. The runner scores, but the batter is thrown out at third trying to stretch a double into a triple. Since the runner on third crossed home plate before the batter was out at third, the run counts. On a force play (at any base) the runner would not be allowed to score even if he crosses the plate before the force is made.
Yes.
It would be considered a fair ball.
Well No, but remember if he swings its a strike The pitch is legal but it cannot be called a strike unless the batter swings and misses or hits the ball foul. The batter may swing at a ball that hits the ground before home plate and if he hits it fair play goes on. In other words, if a batter were to hit a home run on a pitched ball that hit the ground before home plate, it would be ruled a home run.
Yes, it is scored as a home run.
When a batter hits a ball very hard usually in front of home plate that hits the ground and bounces up high.
Home plate is foul territory unless the ball rolls in front of the plate and stays fair. If the ball bounces off the plate and strikes the batter; it's a dead ball.
Yes, either you swing or you don't, there is no middle ground.
Yes. On a tag play, if a runner crosses home plate before another runner is tagged for the third out the run counts.
The umpire. If the runner crosses the plate before a tag for the 3rd out is applied, then the run scores. On a force out, no run scores regardless of the timing.