I've been so busy I haven't had time to get back to this.
I spent a LOT of time I didn't have dealing with this, then, about two days ago, I tried to go back to the original system for DNS/DHCP and suddenly none of the computers could communicate at all – not even by IP address!
I pulled all the CAT5 from the main switch, except for the connection between my iMac and my firewall and they started talking and DNS was working. Then I added one more computer, it worked, and I added the others in steps and now it's working.
Do systems in a LAN talk with each other or cache information that others can get? Because now it's working perfectly, but it took pulling almost every system off the LAN to make it work.