Multiuser - can process of background user kept running?
I'm confident your premise is false.  When another user logs in to Mac OS X using the fast user switching, the GUI processes of other users are not suspended, but do continue to run in the background.  As long as you don't log out of the other accounts, their Finder process and programs launched as that user should continue to run in the background.

For instance, as I type this with my primary cwrea account on my Mac mini, I have another sampleuser account logged in (at the same time), and that user has a Safari window open that is playing a YouTube video.  Right now.

Then, if I load Activity Monitor while having my account cwrea in the foreground and select to view "Other User Processes", Activity Monitor is reporting consistent ~8% CPU usage by a specific process named "PluginProcess" owned by the sampleuser account.

Moreover, if I switch back to sampleuser's login session via fast user switching, the video is at a time location beyond where I left it, consistent with where it would be if I hadn't switched to my main account in the interim.

For what it's worth, the sound (and video, obviously) of the other background login session are suppressed, in that I don't hear or see either, but I can't find any evidence that graphical programs necessarily are suspended or cease operating in the background.