Is it legal to have a shotgun without a permit in North Carolina?
No. I don't think it's legal in any state in America. Look it up dude!
Yes, North Carolina will recognise an out of state learner's permit. The operator will be subject to both the restrictions of the state which issued their permit, as well as the restrictions which North Carolina places on drivers operating on a learner's permit. South Carolina, however, is one of the states which does NOT recognise an out-of-state learner's permit.
There is no limit. However, North Carolina has a handgun permit system which allows you to purchase ten handgun permits annually. If you have a North Carolina issued concealed carry permit, you can bypass the handgun permit system, as the background check which accompanies the application for a North Carolina CCW is one of the most extensive in the country.