I'm going to handle this the same as any other situation when a player is injured during a live ball.
If the injury does not require immediate attention, I let the play finish, call time, and have the injured players attended to. If the injury requires immediate attention for the injured player, I call time, get help to the player, and place runners accordingly.
Specific answers to your situations:
a) Call dead ball when the ball enters the dugout, award R3 home and B4 third. Allow appropriate personnel to attend to the injured F1. I'd probably also try to grab the broken bat and take a quick look at it to see if it had been potentially altered.
b) see A, with the exception of inspecting the bat
c) This one might be a bit trickier. If the injured F2 requires immediate attention, kill the play as described above. If she does not require immediate attention, but is down on or around home plate and there is a possibility of further injury from runners attempting to score, kill the play and award bases.
This doesn't seem too hard. Are you thinking that there is some difference because it was the bat that struck a defensive player as opposed to the ball?
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