The bookmarks on the home screens of my Galaxy Nexus have the same red bookmark icon.
Is it possible to automatically display the site's default icon (i.e., the one which appears in the URL bar) as the bookmark's icon?
The icon that is displayed on the action bar is called a favicon. It's a low resolution (usually 16x16) icon made to be displayed next to the name in a bookmark bar.
Android understandably doesn't use this as it would look hideous on your homescreen. For a website's icon to show up they need to add another icon: the apple-touch-icon. This one was meant to be used as an icon on the homescreen. Originally for iDevices, but android adapted it. Unfortunately I'm not aware of a way to add an icon yourself if the site doesn't supply it.