Hmmm. Let's switch this up a bit.

A is incorrectly given an AP throw in that should have gone to B. Before the pass is released, a) B1 commits a plane violation, or b) A2 fouls B2, or c) A1 reaches the ball across the plane and B1 ties the ball up.

You then realize the error.

In which, if any, situation(s) can you correct the error?

In all three cases, the throw-in was never complete. So, I have...
a) DOG, ball goes back to B.
b) team control foul on A2, ball goes back to B
c) ball goes back to B (the held ball doesn't change the arrow, as the throw-in never completed)

Some of this seems a bit counter-intuitive, but that's what I have.

What if B was entitled to the arrow with the held ball?

B is entitled to arrow and he said B would get the throw-in.

The scenario was that A had the ball for an AP arrow throw-in, and the ball not released yet ... did I have a brain fart?

Right, but I believe PG is asking whether B would keep the arrow. It's a foul on A, so B gets the ball, and it's not an APTI.

In a) and c), however, I have the arrow switching after a completed throw-in.

In a) , did the D.O.G. warning cause the throw-in to end? The throw-in team did not violate.

Read my insert to your quote. :)

This one is making my head hurt.

What did we decide was the cut off for that to be correctible? Was it at the end of the first throw in? Or was it prior to the ball being put in play. If it’s the former I’d argue that all three of those things ends the first throw in, and necessitates a second throw in (or potentially shots if B is in the bonus).

Agreed ... but we still have a held ball on an (incorrect) AP throw-in. Was the whistle blown for a held ball situation? If so, it determines what we have next.

On a held ball of this type (thrower sticks ball through plane) during an AP throw-in the next throw-in is an AP throw-in for the team which had the arrow. Team B had the arrow, but Team A was incorrectly given the throw-in. Therefore it is an AP throw-in for Team B.

This is a good point, as it never was A's throw-in in the first place. Perhaps there's no DOG here.

Okay, so this opens up a can of worms. If you're B, and A has the throw-in incorrectly, should you just step over the line and cause a whistle, thereby getting the ball back when the mistake is realized (and no DOG)?

There is a rule for when a throw in ends. None of these situations qualify.

I'm giving B both.