1. A shotgun wedding is a wedding that has to take place quickly, often because the woman is pregnant. The cartoon implies that Obama was forced to come out in favor of gay marriage because of Biden's remarks, which is why Biden is portrayed with a shotgun for a mouth. The cartoonist may also have been thinking of the expression to shoot one's mouth off, which means "to talk too much in a loud and uncontrolled way".
2. If you are uncertain about the difference between the words 'marriage' and 'wedding', a good rule of thumb is that marriage refers to the institution and wedding to the ceremony.
• The wedding cost over $25,000 but their marriage lasted just a few months.