How Far Will The Networking Neighborhood Take Us?

Is there any possible way to control what someone else visits on the Internet, or change their Background on their Desktop?

Change desktop stuff, yes, and control various things they can or can't do through the local security policies assuming you are using Win2K or XP machines and you are the admin and they are not. Unless you're running a Windows server, however, you'll have to set that at the local computer. Through a Windows server you can do it through group policies.

As far as what they surf - not without a third party cyber-nanny type program.

My school does personalized ColdFusion backgrounds that you can refresh and change (only five times though). The teachers all get their own backgrounds and a lot more stuff on it then we do (text).