Hi Turtle,
Great you got the stone out.
I also use a surgical blade sometimes but for these stones and thin silver bezels I think they are a bit to sharp. Ofcourse this can be helped
But still for these kind of settings with very soft stones a piece of silver or silver solder rolled hard, seems to not damage the stone and leave the bezel intact.
For tougher bezels and harders stones I made a high polished bezel opener that are like surgical knifes but a bit thicker to be able to pry the bezel of the stone.
For your damaged turqoois, you could mix 2component glue (or preferably uv hardening stuff) with some turqoois filings to fill up the gap. Not the best way ofcourse but sometimes a good way to keep the same stone workable.