1) Is the DHCP configured to provide enough IP addresses for all devices on the network?
Yes, plenty. Right now there are only two phone devices connected by DHCP. Everything else is hard-wired.
2) Do you have a USB wireless adapter to test if the laptop can connect with a different wireless card?
Unfortunately, no. I had thought about that, but don't have the device.
3) Can you run a Linux live-DVD/USB on that laptop to see if it can connect that way?
Interesting idea. I don't have such a USB at the moment, but I can download an image and burn it to a thumb drive. If I get time, I'll try that over the weekend and see what happens.
The thing that puzzles me most is that it refuses to even see the SSID for the cable modem/router. I tried manually entering the SSID but that didn't "train" it to see the SSID. One thought I had was that years ago and a few cable modems back, when they first integrated the router technology, I was still using the Netgear router and may have told the Lenovo to "forget" the modem's SSID since it was not being used. (I'm just guessing here, since it was so long ago.) Is there a way to not just forget but to hide a SSID broadcast? I read many articles that seemed to possibly be discussing this, but ran across nothing that was of any help. I've poked through the registry looking for clues, but never found anything useful. There are a bunch of old network profiles in the registry, but I'm not comfortable just deleting them there without creating new residual problems.
I wish there was a way to restore networking on the OS to factory settings, but I haven't seen any mention of that. And without a DVD, an Ethernet or WiFi connection, I have no way to communicate with this machine short of a thumb drive.