I'm not 100% sure about LaCage but i suspect that is one of those theatres where the stage door is on the next street (behind) from where the main entrance is.
As for Phantom, the stage door is in some alley on 45th ( i think between the last theatre and the building that ends up on 8th ave) it's an alleyway that has stage doors for like 3 or 4 of those theatres, I;ve never dont it because I think it is somewhat creepy to go into an alley and wait for people, lol. but i've always wanted to.
http://www.flickr.com/photos/27199361@N08/ Phantom at the Royal Empire Theatre