[dash_riprock]

Originally Posted by Durham

In your first one B is correct, but I believe A is also correct as per the following:

SECTION 11. A batter is out when:
a. On appeal by the opposing team, the individual fails to bat in the proper
turn and another batter completes a time at bat in that place;
(1) The proper batter may replace the improper batter before the batter
becomes a runner or is put out, and any balls and strikes shall be
counted against the proper batter.
(2) If the improper batter becomes a base runner or is put out and an
appeal is made to the umpire-in-chief before a pitch to the next
batter of either team, or a play or attempted play, the proper batter is
declared out and all runners return to bases held before action by the
improper batter. However, any advances by a runner(s), (e.g., stolen
base, balk, wild pitch, passed ball) while the improper batter is at bat
are legal. If the proper batter is declared out, the next person in the
lineup shall be the batter.

Food for thought.

In the second one A and D both seem correct. The run scores and the defense can not appeal. Probably D though.

A can't be correct for the first one. The "next half inning" begins at the time of the 3rd out. The defense has not yet lost its right to appeal the BOO.

On the second one, A is not correct because the run is not an awarded run.

Just my take on it.