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I want to change the background color **inside** the button. How?

As Digg specifies here, only the background around the button is customizable. So until the folks at Digg change their mind, I don't see how.

I want to customize the margin/padding/style of the div that wraps the button. How?

The button is wrapped in a div with "dtom" class. So just add .dtom into your style.css and modify it.