There are two leagues in American major league baseball, the American League and the National League. The American League has the designated hitter rule which allows for someone other than the pitcher to bat in the pitcher's place. The pitcher does not bat in the American League. The National League does not have the designated hitter rule. The pitcher does bat in the National League.
No, once you leave the game you cannot re-enter it, regardless of the positions.
Not in minor league or major league baseball. Once a player is taken out of the game they may not return.
Satchel Paige at age 47 in 1953.
Major League Baseball Game of the Week was created in 1996.
A starting pitcher who is removed cannot re-enter in either league, if he is removed from the game. It's rarely if ever done in Major League baseball, but if a pitcher is removed as a pitcher but placed in another position he can later return to the pitcher position.