"Down a shot and a beer." This is a drink commonly known as a "boilermaker" - you drink a shotglass of whiskey, followed by several sips of beer. (Traditionally, if you're drinking boilermakers, you'll have several shots of whiskey per mug of beer.) By the way, boilermakers are pretty much a working man's drink after a hard day of labor, so Mrs. Clinton was most likely trying to show her affinity for the blue collar worker.
Riding shotgun means you're sitting in the front passenger seat of a vehicle. The phrase dates back to times when the driver of a carriage had to hold the reins and control the horses, and the passenger next to him carried a weapon and fended off attackers.
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