I got a pop-up telling me that there was an update (or new version) to Avast. I clicked on the download button and got a message from Avast that there was a “suspect” item trying to enter my computer. Why would Avast show an update to Avast as “suspect”? I did not proceed.
You can download the latest version here: https://forum.avast.com/index.php?topic=151290.0
That doesn’t sound right… do you remember what exactly the message said? Was there a detection name or something like that?
Sorry. Don’t remember the exact message. Bells were ringing and I got scared and just left the site. I’m not an expert.
I had that when I updated it was hardened mode - aggressive 
Interesting. Well, I guess it shouldn’t block itself, though. 
I must admit I have a little titter when I see it
Since this topic is named so broadly, I’d like to ask why my Avast 2014.9.0.2018 installation doesn’t notify me of the new version? If go to settings->update, I can see a “new version of the program (2014.9.0.2021) is available” text. But where are the systray notfications? I have “ask when an update is available” selected. But nothing pops up. 
It is because the program updates are spread over an update cycle of a few weeks to balance server load. With well over 200+ million user they all can’t update at the same time.
Ok, I totally understand that, but still, if go to settings->update, I can see a “new version of the program (2014.9.0.2021) is available” text. My Avast is ready to install the update, BUT, it’s holding back notifying me in a “visible” way? I.e. if I were to click that “update” button in the “settings->update->program”, I’d be updated? But Avast holds back the systray notification of an available update? I hope I’m able to explain this correctly.
1.-3.: Yes.