Alright, now this is just plain weird. I did some more testing. Here's what I found.
I configured the router to not do DNS relay and I did not fill in the static DNS addresses (so that when a DHCP client requested an address, it should be given my ISP's DNS server addresses). Then I forced a DHCP renew on one of my PC's. Indeed, it got two DNS addresses pointing to my ISP's DNS servers as expected. However, I was unable to do any DNS lookups. nslookup told me I had an unknown server and all DNS lookups I had it try kept timing out. Ping could not ping anything by name. My browser could not get to any sites.
So, then I reconfigured the router to do DNS relay (again leaving the stastic DNS server fields blank). Forced a DHCP renew on the PC. Voila, everything worked again (as expected).
But *then* I got the idea to use the Windows network configuration dialogs to change the DNS server options for my nic to NOT get DNS server info dynamically. Rather, I put the very same two DNS server addresses from my ISP in there (the same two addresses that when previously handed out by the router when DNS relay was off, did not work). I then did ipconfig /release and ipconfig /renew. Now ipconfig /all reported the same values it had previously when things were not working, but NOW everything worked fine.
Huh?
To summarize, when DNS relay is off and the router gives my PC DNS server addresses, I cannot get DNS resolution to work (regardless of whether its giving the PC the ISP's servers, or both the static addresses and the ISP's servers (which I still maintain is incorrect, but now clearly not the root cause of the problem)). When DNS relay is on, everything works.
... and by the way, when DNS is failing, it is happening on all the devices on my network, not just my Windows PC's (e.g. my Linux-based work laptop, my wife's iPad and iPhone, my Nexus 7, and my Galaxy S3 all stop being able to resolve addresses). The second I turn DNS relay back on and the devices renew their DHCP address, everything works again.
So come on now, don't tell me this is working as designed!
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bc