Thanks for all your tips and ideas about where the problem might be but I'm still stuck without any working solution and I did everything that was suggested to me, also provided as much information as posible.
I tried few other things. Firstly I turned my firewall off, rebooted and tried to register my avast installation but no success. Then I restored firewall settings to default, still nothing. Allowed avast in firewall exceptions manually, still nothing. Then I decided to install avast on my laptop which did not have any antivirus installed and I'm experiencing exactly same issues as on this main computer. This computer is connected to my modem by cable while laptop is using wifi. There is no way that the problem is with my computer. Main PC is running win 7 ultimate 64bit, laptot is running win 7 ultimate 32bit and I did complete installation of avast with all tools included on laptot and custom installation with only shields on main PC jus to see if the problem is not some missing tool that I did not include in the installation.
There are 2 things that I can think of as being the problem. One is something is terribly wrong with avast program since few days ago. Other one is that there is something wrong with my internet connection (not sure if its provider or modem) but I'm not experiencing any other internet problems. I'm downloading, online gaming, streaming and browsing the internet just fine, only avast is having some issues. Not to mention that both installations on each PC took several minutes just waiting for that avast_self_extract process to start the installation. Installation was very quick but the wait before I can start the installation is ridiculously long on both PCs. Not sure what avast does in that time since it takes only few seconds for it to create some temp folder in documents and after that the process just hangs and nothing is happening, after several minutes the installation window appears.
Any help appreciated, I'm getting very frustrated with this issue, really. I already sent a package and posted in a diferent topic about how avast slows down file handling which also happens to be problem for both of my pcs and also for other users as seen in that topic and I still did not get any new info.