It depends. If you get one that uses a bluetooth capable MCU (like a nice!nano) then yes generally, although last I looked none of them supported QMK so you'll most likely be forced to use something like ZMK. That could have changed by now though, I know it was being worked out.

Yes, but be aware that you should never plug/unplug the TRRS cable used to connect the two halves while they're powered on. Some Bluetooth models don't use a TRRS cable but instead have both halves connect wirelessly. Also I know sometimes the firmware can be setup to always expect the half that's plugged in to be the right or left half and plugging in the opposite side can sometimes do funky things to the keymap.

classic or cherry I think. Definitely not chocolate, that's exclusively for chocolate switches, not cherry. Also switches aren't hotswap, the sockets are, so there's no such thing as a Gateron hot-swap switch, that's just a Gateron switch.

usually no. If you get a chocolate switch board you need to be careful about the caps, but finding chocolate caps is already a royal PITA. Some exceptionally large or unusual cherry caps could cause problems but that would be true of any keyboard not just a corne. One thing to be aware of is you're going to want a ortholinear cap set, not the standard one as you'll want square modifier keys and things like enter/space that are normally rectangular.