Does there exist a positive integer N such that for all odd primes p the multiplicative order of 2 mod p is strictly less than the multiplicative order of 2 mod p^N ? Once again, references would be greatly appreciated as this pertains to ongoing graduate research.
I don't think this is known. You might want to have a look at:
A. Granville, Refining the conditions on the Fermat quotient, Mathematical Proceedings of the Cambridge Philosophical Society, 98 (1985) 5-8.
You might be able to to say more assuming the ABC conjecture, which may or may not have been proved.