By duplex box I think he means a normal outlet box that has two outlets. He means to check them both and make sure they are on the same circuit. They usually are but they could be on separate circuit in some instances.
As far as permits go, you usually don't have to update stuff to the code that's not directly related to what you got your permit for. For example, if you want to run a new branch circuit for say a hot tub, they aren't going to make you add a ground to all your old Romex wires, or check that your kitchen outlets are only 48" apart. You may not even need a permit for this limited work, but you can find out.