When the router is running as a AP it should not do anything other than convert the packets from wireless to ethernet.
If you are having DNS problems the best test it to not use your router as a proxy and access the DNS directly. You either reconfigure the DHCP to give you a real DNS server or you put the DNS server into your PC manually.
There is a huge difference between a site not resolving and a site that just not load. If you are getting failed to resolve messages it is some kind of DNS issue.
Try the NSLOOKUP command and see if things resolve from a command prompt. This tends to rule out strange browser issues. You can also put entries into the HOSTS file and see what effect if any that has. It will not even attempt to talk to a DNS server if there is a entry in the HOSTS file for a site.