Assuming $P \neq NP$, do we know whether there are problems which are in $NP$, not in $P$ and are not $NP$ complete?
Solution 1:
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.