As far as I know the Avira antivirus has good detection rates and it’s free for personal use, unfortunately I have no experience with it’s firewall. But from what I can see on other forums, avira firewall is easy to configure but falls short on features and reliability..
You can follow this tutorial to make and modify the rules for each application.
I’d recommend getting a better firewall like Comodo, and then uninstalling avira antivir premium and then installing it without the firewall, but you’ll spend a lot of time configuring the firewall to allow the update features of the antivirus. And if you install the defense+ feature of Comodo, the configuration headache gets worse.
If you could do these things before, and you can do them still when you remove/uninstall Avira, then the obvious thing to do is get another antivirus that doesn’t interfere this way. You might try the AVG or PCTOOLS free av products and see if they work better for you.
FWIW, I use Avira with BitComet and have no problems.
I think what may be happening is that BitComet will grab whatever connection resources are available when it starts; if you start your browser while BitComet is already running, the browser may not be able to get access. At least that’s how it seems to work on my system. If I start Firefox first, then BitComet, both work (but they will share download bandwidth).
We haven’t heard back from him for a month. BitComet must be configured, or it will indeed take all the bandwidth it can grab. However, the presence of absence of Avira would not affect this, so it doesn’t account for the reported symptom.