Yes you can have X.org server spawn twice. It's almost one year since I tried to do so, so I don't remember how to do so. I didn't like the outcome, so I reverted my "experiment".
My "experiment" worked well, the game indeed went to a different X.org server instance, and I could use, say, Ctrl + Alt + F8 to go there, and still have my desktop on Ctrl + Alt+ F7. However, spawning another X.org server also takes away some functionalities that one would expect to have, mainly if you're using a keyboard shortcut for some program that is in the background of the first X.org server instance (e.g.: Mumble with push-to-talk) and so you try using that shortcut while in the second instance, which won't do what you would expect. I also face similar issues, but only when recording videos of a game in fullscreen (the game itself plays fine, but the resulting video has dropped FPS). I worked around the problem by running all my games in windowed mode instead, and it even feels better to do so since I can also switch windows with combinations like Alt + Tab (while in fullscreen, some games don't accept Alt + Tab, to the point of not even letting the combination pass to the window manager being used).
