After a couple of weeks, a moderate amount of online research, several phone calls, and endless fiddling, I now have our router configured and our network firing on all cylinders.
We tried several things, me and plesant Indian woman "25219" (I didn't catch her name - something like "Veesha" - but that's her Linksys Technician ID), but the thing that worked was changing the LAN IP range from 192.168.1.x to 192.168.2.x. I can't imagine why that matters, but there it is. It took about 40 minutes. Well, like 3 weeks of poking at it and 40 minutes on the phone.
Anyway, Cindy and I are now both online, and firewalled by the router. Now I'll blow up my hard drive and re-do it to get the spyware and junk off.
For my next trick, I will DHCP my slackware Linux box on and get it Internetable. (You ought to know the rules before you break them - I can get an A in any English class you give me, so I can occasionally throw out words like "Internetable")