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The conventional way to add a page to a book is to glue it along the inside edge. https://en.wikipedia.org/wiki/Tipped-in_page

There are several versions of this article floating around the web showing the process if you happen to have bookbinding equipment laying around https://www.lineco.com/book-repair-3/

There are book binders that will do one offs, and the binding will probably be better than the original if you get it sowed & glued correctly!

Yes, but it is an art. I would recommend taking it to a reputable used book store. They will sometimes have traditional bookbinding skills, if they carry lots of old books.

A library might also have people with the right skills?