There are two separate images and two separate problems posted by the OP, the first relates to being unable to connect to my.avast.com and the second totally unconnected requiring the user to restart after an update. And you continue to link this inability to connect to my.avast.com with other unrelated issues.
Well this is indicating that it is the program update side that is requesting the restart, have you actually restarted ?
I haven’t seen this in regard to this avast7 version, but there used to be an issue some considerable time ago when a registry key (I believe) wasn’t being reset so was still present even after reboot. Unfortunately this was so long ago that I can’t recall the key or element that needed resetting.
Well in general you shouldn’t need to routinely check for Program Updates, as the default settings are set to Ask, so it checks automatically if a program update is available you get a notification pop-up Asking if you want to install it now.
That said, it shouldn’t require a restart unless an update was found and you applied it.
I checked mine, existing program version 7.0.1426 and off it goes to check at the end of that check it reports I’m up to date, I click OK and it is done, no restart required.