Rush, from my experience, is whatever they have left. Since Sound of Music is fairly popular, it may be a bit more difficult to rush but the only time in recent memory I couldn't get a rush ticket was for Newsies. That said, since you're coming from New York and literally have no alternative if you find out it's sold out, I would consider buying in advance. I'm not sure if there are any left but the Paper Mill has very inexpensive rear/side orchestra seats (usually $25-$30 if memory serves correctly) for musicals that have very good views. The mezzanine is very far back behind the orchestra and is miles away. I would definitely grab the rear orchestra if you can get them over rear mezzanine, even if the orchestra is all the way to the side.
The only time I've seen Paper Mill shows on TDF are when the show sells horribly. This is usually for plays and happened often last year during Forum. For Sound of Music, I highly doubt this will be a problem and would bet it will never appear on TDF.
Enjoy and I hope you come to love one of my favorite regional gems as much as I do!
Scratch and claw for every day you're worth!
Make them drag you screaming from life, keep dreaming
You'll live forever here on earth.