I'm going to through some "blind" idea (based on my experience only, and I have no way to know if this is your specific case).
First, delete your firewall rules related to Avast, so they can be rebuilt after reboot.
Double-check your proxy settings (both in IE / Windows, and in Avast). Try testing other settings of the proxy.
Finally, deactivate any "power saving" settings of your network devices (router, network card...). Also, if you have "high performance" settings (either manual or automatic, like 10/100/1000 ethernet for example), disable them too (for example, only 10/100, but don't let the 1000).
After the changes, reboot (and changing the devices properties should request it anyway).
I used to have some little "quirks" with the power saving properties of my network card, and I avoid them by not using those properties.
By no means I mean you shouldn't use them. This is only to try to identify if those properties / settings are somehow related to your problem.