2011 rogue AWD. Has 18" tires. Had a flat. Changed to the temp spare 16".

The different size tires could definitely cause a differential failure, but it is highly unlikely over such a short distance. The sizes you gave are actually wheel diameters, not tire sizes. Are the tires on the car the same size that came on it (check on the drivers side door jamb if you're not sure)? Also, driving tires with more than 2/32nd of an inch of wear difference on an all wheel drive car can cause differential problems.