Well, I formatted my hard drive and started over from scratch. And... it now appears that the problem was in the router. The router worked fine for direct connections, and it made wireless connections look like they were working fine, but they weren't.
I thought it wasn't, before, because I tried bypassing the router and hard wiring the remote computer to the DSL modem, and that didn't work. But now that I think of it, the reason it didn't work is probably because I didn't switch the login protocol over to PPPoE, which is what the DSL modem expects (but is accommodated in the router software).
The semi good news now is that I have replaced the router, and the remote computer is now accessing the internet.
I still have one remaining problem. The network connection isn't recognized when the remote computer first boots up - but if you disable and then re-enable it, it's recognized. Klugey, but it works. If anyone knows how to prevent it, I'm all ears.
Thanks again to all for the help and advice.
Incidentally, I have kept meticulous records of all the steps I went through to rebuild the computer, so I could do it again with my eyes closed...