Just to make some clarifications: In order to switch computers or operating systems you would have to reboot and hence log out of Wow (no instantaneous OS shifting, sorry). However, your character is stored solely on the Wow servers, therefore it does not matter what computer or OS you are using to play, your character is always fine. Your UI configuration and some video settings are stored locally, but they are easily cloned between installations (stored in the WTF folder).
I just recently quit wow, but played almost exclusively on linux (running wine in gentoo) for a little less than a year with no problems. It works well, and I actually got higher max frame rates with linux than windows (I saw less than great frame rates when there was a lot of action though).
I followed:
HOWTO Install and update World Of Warcraft with wine - Gentoo Linux Wiki
To install the game.
Good luck!