No. I'm not sure what a greeting board is, but nyc theatres don't have them. You're there to see a performance, so the only type of board you'll see (potentially) is if one actor or actress is standing in for another at the performance you attend. Go to any of the restaurants nearby if you want birthday wishes.
I doubt there is one, but you can always go to an usher or The Box Office.
I am pretty sure there isn't one but you can always call or ask at the box office.
No, that I know of.
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