Yes ... rule 5.10(c)(1) states:
"If an accident to a runner is such as to prevent him from proceeding to a base to which he is entitled, as on a home run hit out of the playing field, or an award of one or more bases, a substitute runner shall be permitted to complete the play."
A situation similar to this occurred in a game between the Boston Red Sox and Toronto Blue Jays in 2005. Gabe Kapler of the Red Sox was on first base when Tony Graffanino hit a home run. Kapler went down with a ruptured achillies tendon between second and third base. Graffanino waited on second base while Kapler was loaded on a cart and taken off the field. Alejandro Machado entered the game as a pinch runner and finished the trot to home plate in Kapler's absence.
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In the National League, the pitcher is already in the batting lineup. He cannot pinch-hit for anybody else, because he would be batting out of order.
In the American League, the designated hitter bats for the pitcher. If the pitcher was to bat, he would only be allowed to bat for the designated hitter, and that would remove the Designated Hitter rule for the remainder of the game. The rest of the game would have to be played normally, where the pitcher bats in the designated hitter's spot. This would not be a good idea, since most designated hitters are good hitters.
yes. if the runner is on the inner (left side of chalk if you are the runner) part of the base path.