Framing a wall over a carpeted floor
No, I would not recommend nailing through carpet. Whether or not you prefer to, cutting the carpet and pad and removing them (under the wall) is the only right way to do this job. 

Baseboards on top of carpet will look like baseboards on top of carpet. If that's not how the rest of the baseboards are done, they will indeed look "off;" especially in the corners where the two types come together.