I'll try the telnet suggestion and report back. I should mention again though in case it wasn't clear: the router is currently set up to use the ISP's default DNS. Problem stays the same.
As for pinging outside DNS servers, I am pretty sure that when the connection is 'up', I can ping any of the above (e.g. 18.104.22.168, 22.214.171.124) with no problem.
The main thing is that the problem is intermittent, though much worse on Wifi. On the LAN it may work for hours or give the DNS error when attempting to surf for hours. Wifi is like flipping a coin. You might be able to surf three or four pages until it goes out. On the iPad you have to manually reconnect to the network. On the Win7 laptop (when using Wifi) it doesn't seem to want to connect or work at all.