Avast doesn't monitor https traffic.
As I said avast isn't initiating these connections, that would otherwise be down to your browser if the web shield wasn't being used. A single web page can be made up of many, many connections, all the images, links to off-page/site scripts, etc.
I don't know what other connections you want to establish ?
This would appear to be an associate firefox request coming back in through the proxy to your browser:
[firefox.exe]
TCP 127.0.0.1:64751 MyPCName:12080
I have no idea what these are:
[¨t²Î]
TCP 127.0.0.1:65405 MyPCName:12080 ESTABLISHED
Whilst they use port 12080 they aren't directly associated to the avastsvc.exe. Have you got some sort of local network and do you use any other proxy ?
As this would appear to be monitoring local traffic.
This isn't an area I'm familiar with at all as I don't use a network.
Have you changed any of the avast settings in relation to redirection as local communication is meant to be ignored ?