Comment Break this list down into multiple functions (Score 1) 618
Sounds like this list needs to be broken down into multiple sub-functions.
Web filtering, site access control, and total Internet denial are functions for a web proxy or other content filter. You should be able to find a linux-based web proxy that will do what you want in that department.
Scheduling usage hours, forcing logout, etc. is the sort of thing you can do with "policy" objects if you had a Windows Domain Controller. That's probably outside of your budget. But, generally, you need to be looking for client/workstation policy tools.
The computer health monitoring stuff might be part of the policy tools, but might not.