Is there a Firefox feature, setting, or add-on that arranges it so that the favicon of the currently open web page becomes the icon of the browser window in Linux/X11? In particular, it ought to show up in the window manager's task bar. Konqueror does that, and I find it very useful, considering that many "applications" are nowadays web pages.
One option is using Mozilla Prism firefox extension which allows you to run a webpage in its own dedicated window. It will use the favicon for the browser window icon.