Would a studio orchestra prefer the key of F or G?

Aha! I might have the answer.

They just played “Greensleeves” with a scratch on it.

I counted the time for 10 pops to go by and it was 34 seconds. That means 3.4 seconds per rev, which means 17.6 rpm. That means the recording is fast.
It was in Gb (my old friend), so I must assume that the original recording was in F.

From this handy chart:
F4 is 349hz
F#4 is 370hz

349hz * 17.6rpm/16.6rpm = 370hz!

(of course, now I’m wondering if I did something stupid like counting the first “pop” as “1” and now have an off by one error.)

How do transposing instruments work?

I know that they read and finger in a standard system, but the notes may come out pitched up or down, i.e. transposed by various degrees. So, you play in C, and it comes out in various transpostions. Sometimes Eb, sometimes A major, etc.

Is that done by switching to a different version of the instrument, or adjusting the instrument somehow (do french horns still have “crooks”?) or both?

Surely this depends on the instrument. A B♭ clarinet and an A clarinet are two separate instruments, but an electronic synthesizer can trivially transpose any key to any other key.