Odd.
The Internet Gateway thing you saw in your Network Connections area was Universal Plug-n-Play detecting the router on the network (which was UPnP aware).
You can always try the procedure
here, though I don't know if that will help in this situation. Does that Netgear perform any kind of DNS services? From the sounds of it you can connect to the IP's directly, but only when trying to use the DNS names you are having problems. If the router does offer any kind of DNS services, turn them off and see what happens. Also remember that your computer will cache any DNS lookups it does (both successful and failures), so you can have a temporary problem resolving a DNS record, and future attempts to connect to that DNS resource will also fail, even if the real problem has gone away. This will persist until the cache timeout expires, or you manually clear the cache (ipconfig /flushdns).
To be doubly sure everything is cleared, use the "Repair" button on your network connections' status page, as that renews your IP, flushes the DNS cache, the NetBT cache, re-registers your system with DNS, and some other things that flash by too quickly to get a good look at.
Good luck!