Yes, NC requires that the paperwork (or a copy of) be with the NFA item when it is outside of the storage site. It's always a good idea to make a few copies of your paperwork and keep the original in a fire proof safe or safety deposit box. If the original is lost a hassle usually ensues.
This depends on the state. Some states (FL I believe) state that "NFA is legal if it's registered" and that would enable a LEO to legitimately ask to see your federal paperwork. NC may be the same way.
-Mike