Man I wish I had seen the post about just trying to change the DNS settings before I tried to uninstall Pro. I had the same connection error as the OP when I tried to manually update after I got a Windows Action Center notification two days ago that was saying that Avast was reporting itself as out of date. I later logged into my account on the Avast website and found out that Avast had stopped communicating to the servers almost 2 weeks ago.
I tried to repair the install first and that didn't fix the problem. Tried doing a clean uninstall using the same method mentioned in this thread (avastclear, cleaning remaining files, etc.), but now I can't seem to get it to install correctly again.
The "online" installer hangs on the opening splash window and eventually gets an error. The "offline" version installs but doesn't seem to fully install. The desktop shortcut is a generic icon, instead of the normal avast icon for example, although Avast still opens at that point. However, when I try to sign in to my account in the UI it says it can't log me in and it also won't accept my license. I tried typing in the license key and also inserting the license file that i downloaded from my web account. Even after changing DNS settings to Google's servers and re-installing again, I still can't get Avast to recognize my account or license. I feel like Avast is totally borked for me at this point.