Does ginger have to be peeled? If so how? I noticed that many recipes that call for ginger explicitly call for peeled ginger.

When I use ginger I don't bother to peel it. As it is almost always minced fine the tiny bits of papery peel are almost undetectable.

Is there some reason that I should start peeling my ginger? Are there some dishes that are better served by using ginger that has been peeled?

Ginger is knobbier than a potato. Is there a better way to peel them than to just take a vegetable peeler to it?

FWIW- The recipe that provoked this question can be found here.
There's no reason to peel besides the aesthetics of it. There's no major flavor or nutritional difference either way. If the ginger is going to be seen then peel it, if not don't bother. 

The best way to peel ginger is to use the edge of a small spoon to rub it off, it works really well, takes seconds, and doesn't waste as much as using a knife or peeler.