Sounds more like the web shield (avastSvc.exe) is being blocked rather than the other way round. I believe the reason you can get to those other sites is that you are using HTTPS to connect to them.
Over and above Pondus’s questions:
What is your operating system and firewall ?
Does it allow avastSvc.exe internet access ?
- If it does delete the entry for it and reconnect to the internet, this should force the firewall to ask permission again.