I work at the Performing Arts Library, directly with the TOFT Archive. I’ll answer your second question first: No, viewer records are not kept on a permanent basis. They are kept briefly, then discarded.
Next, there is no single list per se of plays & musicals held in the TOFT Archive. Ordinarily, the best way to find out whether TOFT has a particular video is to look it up in the NYPL Catalog. I say “ordinarily” because, unfortunately, there are some technical issues with the database at the moment. It works, but not with 100% accuracy. I’ll provide a link below just the same, but meanwhile, the best way to find out whether TOFT has a particular title is to call our screening room at (212) 870-1642.
Here’s the link to the NYL Catalog, “scoped” to the TOFT collection. You can use the drop-down menu to winnow it to Title, Author, etc. (But again, it’s a little glitch-y at the moment):
https://legacycatalog.nypl.org/search/X |