See this post from a previous thread. It describes how to make Internet Shortcuts on the desktop displaying the website's favicon with IE7:
I don't use Opera, but maybe this will help you figure out how to make it work. The kind of file that you want on your desktop is an Internet Shortcut (.url) file. This type of file has a "Web Document" tab in the Properties sheet that has the "Change Icon" button that Lighthouse mentions. If that's not what opera is creating, you can right-click on the desktop and select New -> Shortcut and paste the URL of the page you want in the location. This will create the proper icon type on the desktop, then you need to figure out where to get the website's favicon as described in the above link so you can change the icon from bland to site-specific.