Nobody ever managed to prove before whether P equals or does not equal NP. As your paper mentioned that the existence of one-way functions opens up a proof to P!=NP, let's discuss that.
if P = NP, then F P = FNP, and so any function that can be computed in polynomial time can be inverted in polynomial time, since there is a simple FNP algorithm that inverts it by nondeterministically enumerating all possible inputs.
However, it is not known whether P = NP implies the existence of one-way functions.
As NotATether said, the "randomness" of parameters should be called "arbitrary".
Yes, in this paper, we can replace "randomness" by "arbitrary" without affecting the correctness of the conclusion.