In mathematical induction we assume $P(k)$ holds for some unspecified value $k$ and try to prove $P(k+1)$. When doing so can we assume that $P(k-x)$ also holds, given that $k-x >$ base case and $x\in\Zeta$?
On one hand this appears valid as $P(k-x)$ is true IFF $P(k)$ is true. However I am worried about the "for some unspecified value $k$" requirement.
Yes, this is sometimes called complete (strong) induction which in itself is a special case of wellfounded induction.
The latter is a theorem (rather a scheme of theorems) in the theory $\mathrm{ZF}$.