Fun game! For question 1, the first person I thought to check on was John Cullum, because his career has been so freaking long. I looked through his Broadway credits and found that he has performed in 20-21* of the currently operating theatres, plus 3 defunct theatres on top of those, making for a total of 23-24 theatres. So that's the new number to beat, and it's a doozy!
*For his Broadway debut in Saint Joan: on IBDB, it's unclear whether Cullum was still in the production when it moved to the O'Neill
For question 2, I wouldn't know where to begin thinking of who might hold that title (for the record, Cullum has performed in the Music Box three times - which, as you said, surely someone should be able to beat).