Is it possible to see two shows in one day — and stagedoor?

I think it's pretty common for actors not to leave the theatre in between the matinee and evening show. When I stagedoored Phantom two falls ago only one lead and one swing came out before I had to leave, and I waited a good thirty minutes. So I'm not sure you'd be missing much if you stagedoored the matinee of MR, especially since the leads are insanely popular — they may want to save themselves the headache of too many fans before having to perform for the second time that day. You might catch some ensemble members if you aren't picky, but I know that the stagedoor in Boston was packed every single time so you may not get much time at all to talk or have them sign your playbill, let alone pictures.

I wish you the best though! I'm actually purposefully going to go to a matinee for MR bc I want to avoid stagedooring it, so you're quite brave!

Absolutely! I do this on almost every trip. If I do not stage door, I find myself having so much time that I don't know what to do with myself. Moulin Rouge will end somewhere in the 4:30-5:00 range. Even if you're at the stage door until 6 (which is unlikely), you'll still have 2 hours before Wicked starts. That's plenty of time to eat and do whatever else you need to do, and even get to Wicked early! Have fun!