My client updates automatically, and silent/gaming mode is active so I don't know when the update came.
Do you mean *definitions* autoupdates, or also the *program* itself?
We have seen firewall issues several times after a program update for avast is performed, even when it seems that the firewall rule is correct. So deleting the relevant rules and re-creating them usually solves this type of problems.
But your case
seems to be different (at least partially), since you are able to connect (for at least a period of time).
Maybe you could try disabling "game mode" in avast for some time, so at least you would be able to see any potential messages? (BTW, do you have the Status Bar options enabled and the pop ups settings too?).
I don't see (until now at least) something in avast that would block a download after some period of time. I mean, either it blocks it, or it doesn't.
You may want to try uninstalling multimedia related tools. For example, uninstall all from adobe (like "flash" and "macromedia shockwave" for example), and be sure to uninstall all versions of them. Then reboot and install the latest stable versions.
I would do the same with JAVA (which can install more than one version in parallel, and they are left behind if you don;t uninstall them manually). JavaRa may be useful for this. So finally you would use only the latest stable JAVA version available.
After rebooting, I would try also updating your web browser, if the problem still presents itself.
Or maybe you should uninstall avast, run the removal utility (several times, each for each version of avast you ever had) and install the latest stable (and reboot)
http://www.avast.com/uninstall-utility.
Maybe someone else has some additional tip / clue / idea.