Have you tried shutting down the Avast providers one at a time?
Is there anything in the system and/or event log?
Have you set a DNS server in the network connection options?

Are you using the latest firmware for the router?
Does giving the systems a static IP changes anything?
Have you tried resetting the router?

Edit: If you use HijackThis, please use the latest version. (Currently 1.99.0)