set theory - Finite axiomatizability and constructible sets

In the development of constructible sets via the Gödel operations, one of the main point seems to be that for $Delta_0$ formulas, one can construct the sets of of elements verifying them with a finite number of Gödel operations, called $G_1$,...,$G_{10}$.

My questions are : does this means that Set theory with separation restricted to $Delta_0$ formulas is finitely axiomatizable since a set is closed under a finite number of these operations iff it is closed under $Delta_0$ formulas? Also since $Delta_0$ formulas are absolute in transitive models, does this mean that if we consider $Th(M)$ with $M$ the fragment generated only by $Delta_0$ formulas, then it is finitely axiomatizable? Now if you had $Delta_1$ formulas is it still finitely axiomatizable? Finally, does this imply that a reflection theorem can't hold in Set Theory with $Delta_0$ separation?

I hope my questions were accurate.