What will happen if disk space is exhausted on a WSUS server

Does the server have to run Windows 2003 Server, or will Windows XP Professional work?

Yes, Windows Server is required.

Where do I download the WSUS packages from?

Microsoft's website: WSUS 3.0

How can I configure my computers to look for updates from my server instead of the internet?

There's a deployment guide & a step-by-step guide that explains it all in detail. The executive summary is:
If your computers are members of a domain, then you can use group policy.
If your computers are stand alone, then you will have to modify the registry of each machine. THis KB covers both scenarios.

It sort of depends on how complicated your group layout is and whether you've done a lot of custom approvals/denials for certain groups. It also depends on whether you care about the history and current status of the computers in the database.

If you have a very simple setup where you're just approving all updates and you don't really care about the history of any of the computers, there's no need to backup the database or the updates. It would probably take less time to just re-build the service from scratch (depending on how fast your network connection is to Microsoft) than restore from backup. All of your existing machines will just check in with the new server once it's up.

It's a pretty basic risk analysis for any service. If you have data that would be costly (time or money) or impossible to re-create from scratch, you should back it up. Otherwise, don't bother.