No, as soon as the batter is hit by the ball, the play is considered dead and the batter gets his base.
They go homeplate
No it is a hits batsmen. The batter gets first base and the ball is ruled dead. Everyone who is forced moves up one base.
The pitcher does not get an assist unless the batter strikes out and the ball bounces to the pitcher, who throws the ball to the catcher, who tags out the runner.
If the ball touches the ground even before the catcher catches it the batter/runner may attempt to advance to first if there are less than 2 out and first is not occupied or if there are 2 out and first is occupied. The catcher does not have to catch the ball on a third a strike for it the be considered a strike out. This includes a ball which bounces to the catcher because the strike zone is over the plate not where it is caught.
The batter or the runner? Either would have to be tagged to result in an out. The catcher (or any player) would have to have the ball and touch the runner from third. If you mean the batter is still in the batter's box and the ball rebounds and hits him then I would surmise that it is a judgment call by the umpire as to whether or not the batter was in the way of the catcher being able to make a play.
The batter can run to first if the 3rd strike isn't caught. However, the batter still can be thrown out by the catcher at first base. Yes. A strike is called if the ball crosses home plate at the correct height, which is usually between the knees and the letters. A pitched ball does not have to be caught by the catcher to be called a strike.
The catcher, and behind the catcher is the Umpire.
When any foot crosses into the box.
Batter's interference is when the batter interferes with the catcher's ability to throw and catch a ball that is in play. An example of this is if the batter were to obstruct the catcher when a teammate is stealing a base, resulting in the catcher's inability to throw out the runner.
The catcher is allowed to touch the batter while throwing to any base. However, the batter must not move into the catcher's path. If the batter does and the catcher makes contact with the batter, it is called as Interference. If Interference is called during a steal attempt, the runner must go back to their original base.
If the batter is in fair territory, the batter is out. If the batter is not in fair territory, for example the batter is still in the batter's box, the ball is called foul.