I'm not sure where my mistake was [1], but I ripped out the DHCP (Dynamic Host Configuration Protocol) settings, added them back in with one small difference (the DHCP settings are tagged with a name, which I originally had as LAN_ACCESS but renamed it 10.10.10.0/24) and that seemed to do the trick—DHCP was working fine under IOS [2] 12.3. Why that fixed things, I don't know. Maybe I knocked some bits loose or something.
Now, testing DHCP and NAT (Network Address Translation) on the Cisco router—that proved to be a bit more challenging. I needed one computer to act as an upstream, and another one to test that DHCP and NAT were working. Getting a few computers wasn't the problem. Getting a few computers that worked without issues was the problem. It was mostly networking issues. One, my Linux laptop, with a supposedly 10/100Mbps (Megabits per second) interface, refused to link to the 10Mbps port on the router. And the Windows laptop … well … yeah, it's Windows, running a few different firewalls that all needed to be shut off, and extraneous interfaces (such as the wireless interface) shut off and …
Ick.
Anyway, router is good to go.
I hope.
Also, I talked to Smirk about the non-automatic routing changes that have to happen at F. Smirk's answer to that? If that's the way it works, that's the way it works.
Oh well.
I can only hope that the problems F is experiencing are due to an overloaded consumer grade router and that once this Cisco goes into place, there won't be any more issues.