Each runner, other than the batter, may without liability to be put out, advance one base when there is a balk or the batters advance without liability to be put out forces the runner to vacate his base, or when the batter hits a fair ball that touches another runner or the umpire before such ball has been touched by, or has passed a fielder, if the runner is forced to advance
The bases of the Baseball diamond are 90 ft apart. The diagonal distance between 1st and 3rd bases is therefore about 127.3 ft (90 feet times the square root of 2). The pitcher's mound lies (roughly) in the center so it will be half the distance between 1st and 3rd bases - 62.6 feet. That's how far the pitcher has to throw to get a base runner out on 1st. It's the same distance he has to throw to get the ball to the catcher (roughly - since he steps toward home plate as he delivers the pitch).
The official scorer would have to determine if the throw was catchable. If the throw was, then the error would be on the first baseman and he would be charged as such. If on the other hand the throw was a bad one and the first baseman had to reach and could not catch the ball, then the error is on the pitcher. Only one error would be charged even though the base runner advanced two base and scored.
I believe he throws it back to the Pitcher.
No, that is a violation and the batter takes his base (balk)
Yes. But he has to be careful not to balk.
Yes. When there is one or more runners on base, it is mandatory for the pitcher to hold his arms to his side before entering his Setup Stance. When entering Setup Stance, he must hold the baseball with both hands and come to a complete stop before throwing to any base, including pitching. However, if the runner wanted to be stupid and decided to run before the pitcher completes his Setup, he can throw the ball.
Yes because of his errant throw the runner was allowed to move up ,at least, one base
runner is out
A catcher tells the pitcher what kinds of pitches to throw. Also, if the pitcher throws a bad pitch, the catcher can prevent the keep the ball close to prevent any base runner advancements.
Yes, the runner would go back to 1st base if the batter fouls off the pitch.
That depends on where the 1st baseman fields the ball. If the 1st baseman can field the ball, throw to second, and have enough time to get safely back to the base to catch the return throw, then (s)he will cover. If not, then it is the responsibility of the pitcher.
Error on the throw
Yes, but he must first make a bluff to a runner on third--then bluff or throw to first.