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How can I change the image displayed by the widget?

Simply open the widget option page (Design > Widgets) and select one from the dropdown list.

Is there a place where I can see all of the badge images displayed by the widget?

Visit this page for a complete list of the badges: http://www.mindsharestudios.com/barack_badges/

This thing is messing up my sidebar! How can I fix it?

While the widget should work with most themes out-of-the-box, you can customize the way it is displayed easily using CSS or by simmply adding some HTML or inline styles using the widget options page (see the screenshots).