Sounds like a great trip! I'm not sure I would do the switching thing. I think it would be fine most of the time, but some people at the stage door can be weird, and if someone was nasty about it, it might ruin the whole thing for you. Also, if the stage door is really crowded, logistically, it is not realistic. Fortunately, none of the shows you have listed have really crowded stage doors, and it shouldn't be necessary to reserve a spot for your daughter. Being in the second or third row isn't usually a problem in terms of getting the actors' attention -- just requires some reaching.
War Horse: As already mentioned, there is no need to switch for this show -- when I went, there were literally two other people there. The cast is extremely friendly and nice.
Mormon (matinee): This is the only one that might have a large-ish crowd, but when I saw it at a matinee in May, the crowd was pretty small - I'm not even sure they put up barricades.
Phantom: Ask them where the actors come out; if it's at the stage door, it's across the street
Wicked (matinee): If this is a two-show day for them, the principals might not come out
Billy Elliot: The adults don't all go down the barricade, but Billy will
Hair: cast very nice
I think the big thing to do beforehand if you have not already done so is to manage expectations for her, particularly for the matinee shows on 2-show days. Some of the actors will come out before she gets there, and matinees are very hit or miss. For example, I saw Wicked and the only cast members of the 7 principals to come out were those playing Madame Morrible and Boq; I saw Follies and none of the four leads came out; I saw Hair and both the actors playing Claude and Berger did not come out.
As an aside, I have also noticed that Lincoln Center shows do not have big stage door crowds, although I am not sure I understand why. At Coast of Utopia, I was the only person. At South Pacific even with the original cast, there were maybe 5 - 10 people. I agree with what has been said that it is much more relaxed and nicer for everyone with the small crowds.
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