AvastUI.exe is the gui and runs the tray icon, kill that and you kill the tray icon and the GUI (User interface)
AvastSvc.exe is the core engine in Avast
the two work in tandem
if you look in your firewall rule under "PROTOCOL" you set "BLOCK ALL" , this means block all means of communication - the name is just the name for that rule, if you changed it to GRC.com and internet miracously started working then the rule must have been disabled or changed...simple as that!
the reason your internet doesnt work when you block AvastUI is kind of self-answering...
A) Avast might think you have a virus trying to block it, so disables internet connection incase its trying to 'bot' your machine (unlikely)
B) If you block Avast through its own firewall then your essentially causing problems with its functionality, as it 'controls' your web traffic through the firewall, network shield and web shield then doing this is bound to cause problems! (likely)