Considering that you can make saxes with lacquer coats, silver plate, or even make them out of plastic, it is hard to imagine that something as subtle as altering the crystal size in the metal would make an audible difference in a wind instrument.
When I first heard about cryo-treating a sax, I mentioned that there were a number of finishes I'd have a hard time wanting to treat because they might get ruined. These were lacquer and enamel. The former is what, oh, 80% of saxophones are finished with.
I also hesitate on any plated finishes. I don't know how the bonding of materials (silver + brass, gold + silver + brass, etc.) would be affected by a cryo treatment, if at all.
Strings, yes: all of the sound comes directly from the string's vibration.
Yes and no. I would say that the strings
start the sound. The body of a violin, say, is what amplifies and colors the sound.
As an example, I've heard violins that don't have a body. It's a different, quieter sound.
For woodwinds, at most a negligible part of the sound comes from vibration of the body
This is debatable, to a certain extent. In the flute world, there have been studies that suggest wall thickness and the
amount of tone holes actually makes a difference in the sound (I have the articles in a different thread in the Flute area).
However, flute != sax != clarinet. There's something to be said for the bore.
IMO, cryo has always been snake oil.