You may be able to ping 126.96.36.199, but is your router set to use the 188.8.131.52 DNS service (or any other valid service)?
Try opening a terminal on your Debian machine and changing to root. Then use the command:
and see if it finds Googles server or not.
If not, note where it stops responding and try:
and you should get a final response;
If you CAN traceroute 184.108.40.206, but CAN'T traceroute google.com, there's a high probability it's a DNS configuration problem your end.
Note: If you can't change to root, then simply place sudo in front of each command, but you'll still need the root password to run the command. It may actually be possible to run traceroute as user, but that will depend on how your machine is set-up.
The Chrome issue is known, and is specific to Chrome. So I suspect it has very little to do with your other connection issue. Once you can connect normally, try clearing the cache in Chrome, That may help.