First thing

Disable MBAM. You don't need that running in the background all the time as it might conflict with Avast. Just run it once a week or so to update it and run a manual scan. Reboot after you have disabled it and see if that fixes the problem.

Next, firewall. I'm confused. You only just turned on the Windows Firewall? Does this version of Avast have a firewall included? If so, turn off the Windows Firewall. You don't need both.

If Avast does not have its own firewall, then as a test, turn off the Windows firewall anyway and see if that solves the problem. Don't forget to turn it back on if that is all you have.

Norton.Did you uninstall Norton before or after installing Avast? Norton is known for causing problems when its license has expired so you would do well to download and run Norton's own Uninstaller from here;
https://www-secure.symantec.com/norton-support/jsp/help-solutions.jsp?lg=english&ct=united+states&docid=20080710133834EN&product=home&version=1&pvid=f-home

If you still can't download anything, get that file downloaded on another machine and then transfer it across. It may be that Norton's own firewall is still working in the background.

See if all that helps.

Mark