What's the best way to store unused sushi rice?
Storing cooked rice is generally not a good idea.

With the high starch content in rice, it is simply an excellent growing ground for bacteria. Add to this the fact that uncooked rice can contain spores of Bacillus Cereus which when germinated to bacteria can lead to food poisoning causing vomiting and diarrhea.

Bacillus Cereus spores can survive cooking, so when rice is left to cool they can germinate and start multiplying in the rice. It isn't the bacteria itself that is dangerous, but a toxin they produce. This toxin does not disappear from re-heating.

If you do want to store the rice, make sure to cool it really quickly and store it in the fridge for no more than a day. Personally I never do that and only cook as much rice as I need.

Anecdotal: In fact, a friend working at a big sushi chain in London told me recently that over the course of four or five years they had one incident of proven food poisoning in one of their many restaurants. The source of the food poisoning was not the fish, but actually the rice which had been standing too long.

Storing nori I don't know much about, but if you seal it in an airtight bag and keep it dry I see no problems with that.