All bets off on the toasted board. No telling what voltages its sending on the 5V/3.3V lines, so it could cook many things.
Your PSU as 12.27V shoukd be fine for a good board. Shame about the toasted driver.
For your questions:
yes, I have my SKR Pro powered up with no motors connected right now.
yes, mine is powered with the driver power and board power in parallel from the same supply.
I think the new board should be able to protect itself from damaged drivers. The smoked one should be discarded.
i dont think that the driver needs to be the same brand. The BTT ones are Chinese manufacture, too. They are the same driver, with the same polulo pin configuration, I think the local replacements will be fine. A heat sink may be a good idea, you’ll still be using it for the same duty cycle and current settings.