BOT question
 

R2 on 2B, no out. B1 bats out of turn. The defense appeals the BOT to the umpire after B1 reaches first. In each case, does R2 return to second base if

a) He runs to third on a third strike wild pitch?
b) He runs to third on a ball four wild pitch?
c) He steals third on ball four, not wild?

Would it make any difference in cases b or c if R2 were on 1B and ran to second on the pitch?

Another question: In BOT situations, should the umpire ever "help" the offensive team in determining who is the proper batter (even if they ask?)

If the runners advance was NOT as a result of the batter's actions then he keeps the advance.

BOO Question
 

Only if the offense asks.

In each case, does R2 return to second base if:
a) He runs to third on a third strike wild pitch? NO
b) He runs to third on a ball four wild pitch? NO (unless 1st base was occupied)
c) He steals third on ball four, not wild? NO (unless 1st base was occupied)

Would it make any difference in cases b or c if R2 were on 1B and ran to second on the pitch? Yes. since his advance was technically forced due to the batter becoming a runner, his advance would be nullified upon appeal.

MLB 6.07(b) When an improper batter becomes a runner or is put out, and the defensive team appeals to the umpire before the first pitch to the next batter of either team, or before any play or attempted play, the umpire shall (1) declare the proper batter out; and (2) nullify any advance or score made because of a ball batted by the improper batter or because of the improper batter’s advance to first base on a hit, an error, a base on balls, a hit batter or otherwise.
NOTE: If a runner advances, while the improper batter is at bat, on a stolen base, balk, wild pitch or passed ball, such advance is legal.

Correct. I was referring only to scenarios B & C in which the improper BR recieved a base on balls.

B3 bats instead (and either strikes out or gets a single) of the proper batter B2. A pitch is thrown to the next batter which is B2. The defense appeals BOOT and you tell them that B2 is now the proper batter. B2 finishes his at at bat with either an out or a hit. The offense asks who is the next batter. Do you actually tell them B4 or B3 depending on the sit?

When the proper batter is on base, he is passed over, and the following batter becomes the proper batter.

Rule 6.07 PLAY 6. Daniel walks and Abel comes to bat. Daniel was an improper batter, and if an appeal is made before the first pitch to Abel, Abel is out, Daniel is removed from base, and Baker is the proper batter. There is no appeal, and a pitch is made to Abel. Daniel’s walk is now legalized, and Edward thereby becomes the proper batter. Edward can replace Abel at any time before Abel is put out or becomes a runner. He does not do so. Abel flies out, and Baker comes to bat. Abel was an improper batter, and if an appeal is made before the first pitch to Baker, Edward is out, and the proper batter is Frank. There is no appeal, and a pitch is made to Baker. Abel’s out is now legalized, and the proper batter is Baker. Baker walks. Charles is the proper batter. Charles flies out. Now Daniel is the proper batter, but he is on second base. Who is the proper batter? RULING: The proper batter is Edward. When the proper batter is on base, he is passed over, and the following batter becomes the proper batter.

I do.

They don't bat out of turn. They bat out of order. BOO

OBR 6.07 Batting Out Of Turn

I think I screwed that up. When a pitch was thrown to B2 it legalized B3. So if the defense appeals B2 BOOT then I should swap B2 for B4. If the defense appeals B3 BOOT, let's say if he got a single, I tell them he is legal now that a pitch was thrown. But do we have to have a specific appeal about B2 to swap him out for B4 in this case?

First you had the offense asking, now you have the defense asking.

In general, I'd try to fix the problem entirely once a team says something. In the pros, it might be different.

I had a case several years ago in a Legion game. I was BU. Offensive coach comes out after 1st batter of the inning grounds out. After brief discussion my partner signals 2 outs. I call time to have a discussion with my partner. Seems he got confused. The batter who grounded out was batting out of order, 5th batter in lineup when 4th should have been up. So he was signaling one out for the groundout and one out for the BOO. I tell him no way we can have one batter and two outs. Defensive coach comes out to join the discussion. We tell him we have a BOO situation. He says he is happy with the result and he did not bring it up. Skipping #4 and #5 grounding out seemed to him like a good deal. I tell my partner that since defensive coach did not bring it up, offensive coach did, we we have nothing to enforce.

My point is, if offense asks that don't mean much, unless they ask before the at bat is over.

6.07 BATTING OUT OF TURN.(a) A batter shall be called out, on appeal, when he fails to bat in his proper turn,...

Rule 2.00 (Definitions of Terms). An APPEAL is the act of a fielder in claiming violation of the rules by the offensive team.

Translation: If the fielding team doesn't bring it up, ignore it.

FWIW BOOT can be appealed by anyone on the defensive team. Doesn't have to be a fielder.