Linux, minus the freedom?

PClinuxOS will be great for this.

Once you get, uninstall the few games it comes with.

It has a GUI of parental controls that will enable you to configure all of those options for each individual user account.

It has great hardware detection and pretty much works flawlessly out of the box on any computer I install it on.

That's all do-able, but the scope of this goes way beyond what Yahoo Answers is meant to deal with. You would essentially do an install of something, like Ubuntu, and using the super-user/root account get the software you want on it. Then you'd setup the internet site list (can't remember the file name of the top of my head) to only allow specific IP addy's and addresses. Then you'd set up a script that would start running a timer or cron job when a user logged in. After X time went by, they'd get booted out. That's very, VERY user-unfriendly, though, because someone may be in the middle of doing something important and needs to save their work or such.

Linux (based on Unix) was designed to be locked down and controllable, so it's all do-able. But this is the kind of stuff people sign up for $300 continuing ed classes at community college to learn, though. You should probably post this question in like the Ubuntu or Red Hat forums, and they'd be able to direct you better. Sorry.