I'm trying to understand how the Levy hierarchy does not contradict the axiom schema of restricted separation.
In his paper on the set hierarchy Levy introduced satisfaction predicates which I think implies that for any fixed quantifier rank $k$, there is a predicate $\phi_k$ of the same rank with one additional free variable $x$ such that $\phi_k(x)$ is true precisely when the formula of rank $k$ with Godel number $x$ is also (assuming $x\in V_\omega$ is part of a Godel numbering scheme). Also, since such predicates deal with all formulas of a fixed rank they are not equivalent to formulas of a lesser rank.
Assuming I got that right, then I don't understand how we can have the separation axioms be consistent because we would have: $$\exists b\; \forall x\; x \in b \leftrightarrow \phi_k(x) \wedge x \in V_\omega$$ but the left side of the bi-conditional is just set membership, a formula of rank $0$, so how could it be equivalent to a formula of a much higher rank? Is the issue that I need to account for the set $b$ not being part of the language itself but rather the universe of a model? Thanks.
Yes, that's on the right track.
For example, it's not very elucidating to say every set is definable by an atomic formula just because we can instantiate some variable $b$ standing for some set and taking the formula $x=b.$ (Even though it's technically true, if by "definable" we mean "definable with parameters", so we need to be careful about what type of definability with mean...)
In the case of the Levy hierarchy, we say (for instance) that a formula $\varphi(x)$ is $\Sigma_n$ over some base theory $T$ if there is a (syntactically) $\Sigma_n$ formula $\psi(x)$ such that $T\vdash \forall x(\varphi(x)\leftrightarrow\psi(x)).$ Note there are no parameters here.