We have the result that $ZFCfin$, the usual $ZFC$ axioms with the axiom of infinity replaced by its negation, is bi-interpretable with $PA$, first order Peano Arithmetic. We also know of a natural way to weaken $PA$ into fragments, by restricting the induction axiom schema to forumlae of a specific complexity, so $Q+I\Sigma_3$ is the non inductive axioms of $PA$ plus induction restricted to formulae of at most $\Sigma_3^0$ complexity in the language of first order arithmetic.

Does weakening the axiom of separation and the axiom of replacement in $ZFCfin$ result in the above outlined fragments of PA? For example, does weakening the two axiom schema to formulae of $\Sigma_3$ complexity in the language of set theory give a set theory bi-interpretable with $Q+I\Sigma_3$? Or does the axiom of powerset also need to be dropped and then replacement replaced with collection?

If restricting separation and replacement is the correct way of getting bi-interpretable theories, does a theory bi-interpretable with $Q$, Robinson Arithmetic, result from dropping both axiom schemes entirely?