The only baseline software reason I can determine avastui.exe is used for is the WebRep feature. I don't use that feature.
The avastUI.exe is the graphical interface and is used by many shields, I believe that the alert windows are also handled by the avastUI so if that isn't running, I guess you wouldn't see the alert window. You could test that by downloading the eicar test file whilst you don't have the avastUI running.
I don't believe it is required by the WebRep to display the WebRep information, as I believe that would be done by the browser as essentially it is a pop-up displaying the data when you click on the webrep icon, etc.
I personally detest "cloud" concepts and processing. To me it equates to giving vendors a built-in spyware backdoor; something by the way that MS has built into their OSes since day one. The risks of clould compting far outweight its benefits.
I guess you are going to have a hard time with that one, as it seems that this is the way most AVs are going. I'm no cloud fan as when your internet is down so to is that element, but it rather depends on how heavily the AV is dependant on cloud processing.
As far as my situation goes, I could live with the Avast advertising but not when DNS resolution is to questionable sources.
Also closely look at the WhoIs data from my original screen shot. You will notice that the Indian city mentioned is Bombay. Has that city not been named Mumbai for sometime?
The resolution of the IP address isn't something in the control of avast, that is down to whatever application (TCPView) resolves it and the DNS server it used to resolve the IP address.
Get the IP resolution wrong and the whois details taken from the domain name (resolved IP address) will also be wrong. As Vlk said "Reverse DNS lookup is often bogus." perhaps, bogus should be replaced by wrong.
As you found doing a whois on the wrong domain name returns a different IP address, so the problem is one of incorrect resolution of the IP address...