I don't use dock apps, just KDE panel with task manager. I want to know if it is possible to change the browser task icon to match the site favicon instead of the browser icon?
I use Firefox 4, KDE 4.4 and Ubuntu 10.04
What you are trying to do may be possible to do with a firefox extension (try mozilla addons), but I don't believe this is possible without some complex changes to firefox or a very complex script, as the favicons are saved in your cache and it would be some what difficult to get firefox to dynamically change its icon constantly as you browse.
Maybe you could use prism if there is some specific website that you want to see in the taskbar with an icon.