I have been trying for days to connect to an apple server to update apples QuickTime and not once has it connected to the server, just keeps throwing up the error you are probably getting but the one I get also mentions the server might be busy (probably).
Though I would have though the itunes servers would be more readily available so I don't know if something similar is happening.
I finally gave up and QuickTime is no longer on my system, history.
You could try 'Terminating' not pausing, the Web Shield and Network Shield (answer No about persist changes) as they are the only avast interface with the web and try again. I doubt it will change anything.
Unfortunately I don't use a router so I can't be of any practical help there.
The D-Link router migh well provide a hardware firewall but that probably no outbound protection for that you will ned a software firewall, something to think about for later.
Any malware that manages to get past your defences will have free reign to connect to the internet to either download more of the same, pass your personal data (sensitive or otherwise, user names, passwords, keylogger retrieved data, etc.) or open a backdoor to your computer, so outbound protection is essential.