Well avast doesn't scan https connections as they are secure encrypted, so it shouldn't even be scanning that.
If you are using http port 80 for streaming then this might be the problem, if the stream doesn't completely comply with the HTTP protocol, unfortunately being on dial-up I can't easily test that. I'm using firefox and clicked one of the links under the Webb-TV heading (Prästtomtabor protesterar mot försvarets planer) and that opened up a new window, whilst there is a lot of data transfer going on but no pictures yet. That is most likely the caching to get to a point where has a big enough buffer, so it doesn't look like I can test that.
The new window that pops-up uses a different IP to the normal home page, so you could try adding this IP address (194.14.241.138) to the Web Shield, Customize, Basic tab, Ignored addresses: filed and see if that makes any difference.
From image 2 you can see I just managed to get a little buffered, so it is working for me all be it slowly because of dial-up.