Here's a question. Have there been any theoretical results showing that if P != NP, there must exist some problems in NP which are not NP-complete and which are not in P either? Just curious because I've never seen this question addressed.
Yes, this is known as Ladner's theorem, proved by Richard Ladner in 1975. The class of such problems are called NP-intermediate. Here's one proof that does "cut and paste" between some problem in P and some NP-complete problem, here are two, and there are others.
There are also problems that are already believed to be neither NP-complete nor in P, like integer factorisation and graph isomorphism, as well as problems known to be in NP ∩ coNP but not known to be in P.