Wow. I cannot believe that none of you professionals even had a measly suggestion. Eventually we fixed this problem ourselves, without any experience at all. My faith in forums is poorer now than it has ever been. I hope to fix that by following up on my own success. If anybody out there has a similar problem, try what I did.
Use a computer, like a friend's newer laptop, which has a connection to the internet so that you can download a few things. You will need a USB drive or something like it to move files from the connected computer to the unconnected computer.
Here are some tricks that helped me advance my diagnosis:
I made sure I had the right Belkin (wireless adapter) driver installed. When my computer got buggy with it I uninstalled it and installed it all over again. When the Belkin utility began identifying other networks and attempting to connect to them, I knew that the Belkin was working properly.
Because I'd just reformatted my hard drive I also double checked that my motherboard drivers were installed and working. You can check to see if your devices have errors by clicking in 'control panel', 'system' and the 'hardware' tab at the top, then the 'device manager' button. It might not have been necessary, but there were a couple of USB devices that had a yellow exclamation point next to them. I updated those drivers by going to the motherboard's web page and finding the updates.
I think the thing that finally got me online was installing service pack two (full), service pack three, and something I found on Microsoft's page called microsoft.net framework version 1.1 and subsequent versions.
After you get sp2, sp3, and .net framework version 1.1, try to get online with the windows application, not your wireless utility. Disable your wireless utility by going into it and find in options or wherever it may be to 'use standard windows to configure your internet connection.'
Now go into available wireless networks and click on the network you are trying to access, then go to to change advanced settings, wireless networks, and add the name of the network. Make sure to get the password and the authentication specs right.