BOO & Continuous Action

[Suudy]

Something I'm not clear on with regard to BOO. Take this sit:

R3, 1-1 count, and a) 2 outs, b) 0 or 1 out. On a WP, R3 advances and scores. After R3 scores, B1's coach realizes B1 is BOO. The coach corrects the BOO and sends in the correct batter.

Does the run stand? There's in effect no penalty for BOO if corrected before the at bat completes?

Also, with regard to when the at-bat ends. Say R3, 1-2 count, and a) 2 outs, b) 0 or 1 out. B1 swings and misses, and the ball gets by F2. R3 advances and scores, and B1 is safe at first. Does the at bat end on the third strike as well?