You can use calipers or a good engineering ruler to measure as with wood shrinkage it is unlikely to be a standard size. It is generally safer to go...
I have an inexpensive phone holder in my car that clamps onto the air vent fine. It should work. Just do a web search for "car air vent phone holder"...
To me a 50%0price drop is a big red flag. It could be worth the money but I doubt it. If you decide to sell it down the road you would have a hard go...
I am mostly a bass player but bought an old flatback mandolin in an antique store several years ago. I enjoy playing it and have become more interested in mandolins.