I tried one recipe with silken tofu, but it didn't set enough and just ran off the carrot cake. 
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.  