I think a lager is the most common for the beer batter, but I have heard of people using ale and some want a little more flavor and go with a stout.
The thing about cooking with Coca-Cola, Dr. Pepper and Root Beer probably can be traced back to the war years when sugar was rationed - colas were not rationed and several recipes were adapted to take advantage of the sugar in the drinks. One of the biggies was using cola to baste hams, Root Beer and Dr. Pepper were especially common - the Root Beer for it's sassafras and Dr. Pepper for it's prune flavor. Coca-Cola cake was another big hit during those years.