The play is dead, the runner that was hit is out, and the batter reaches first base and is credited with a single.
If the batter is struck by the pitch in the strike zone, the ball remains live so if it was the third strike it would be an out. The runner could score from third base.
it is scored a ball and the at bat continues
Yes as long as the runner advances Assuming you mean the runner tags up on a fly ball and advances to second, it is not scored as a sacrifice, but, simply as a fly out and the batter is charged with a time at bat. If the batter bunt a ground ball, the runner would not be required to "tag up" to advance and the batter would be credited with a sacrifice. If the batter is attempting a sacrifice bunt and pops up and the runner, tags up and somehow advances to second the batter is not credited with a sacrifice.
If the runner is in fair territory then the runner is out. But if the ball hits the runner in foul territory, then it would just be a foul ball. If the ball hits the runner in fair territory, the runner is out but if the batter is safe to 1st then they would be safe.
Batter is charged with a strike, ball is dead (like a time out). No runner can advance.
The batter is awarded a single. MLB Rule 10.05(5) states that a batter is credited with a base hit when: "A fair ball that has not been touched by a fielder touches a runner or an umpire, unless a runner is called out for having been touched by an Infield Fly, in which case the official scorer shall not score a hit".
The "batter" becomes a "runner" if he is walked, hit by a pitch or puts the ball in play. He becomes dejected if he strikes out, fouls out or is ejected from the game.
The runner is safe, in order for the runner to be considered out the fielder would have to have the ball in his glove or hand and tag the runner. *edit: It depends on where the runner is when he is hit. If the runner is running a direct path to first and is within the 3ft baseline, the runner is safe. If however the runner takes a lane outside the 3ft path, in a deliberate attempt to block the catcher's throwing lane, the runner is out for Interference.
it is scored a wp the batter doesn't get RBI
The batter could be credited with an RBI if, in the scorers judgment the base runner would have scored had there not been any overthrows (errors). For example, the batten singles and the runner is on third and scores, then the overthrows occur. The batter is credited with one RBI. In your original question, the batter would, in no case, be credited with an RBI on his own score.
walk, error, fielders choice, player interfeerence, dropped 3rd strike, pinch runner and hit by pitch 1. Base on Balls 2. Intentional Base on Balls (scored separately to 1) 3. Hit by Pitch 4. Error 5. Fielder's Choice 6. Passed Ball 7. Wild Pitch Pinch Runner is NOT a way of scoring a REACH of 1st base. Runner Interference (if the batter himself is not out) is scored as a base hit. Catcher Interference is scored as an error. Fielder Obstruction is scored as an error. Any scenario you can think of will be scored as one of the above 7, or a base hit. (I am only referring to how the reach of first base is scored)
If the batter showed signs of trying to move out of the way to give the catcher a clear lane to throw then neither the batter or the runner it out. If the batter did not move at all to provide the catcher a throwing lane, then the batter is out, but the runner is safe.