What are the axioms of automata theory?
Automata theory is a broad field and researchers keep inventing new types of automata. There is even a conference series on non-classical models of automata. So it is impossible to encompass all of automata theory. For classical models such as context-free grammars, recently formalized proofs of the most basic theorems have been carried out. These are derived from first principles (these are definitions in terms of set theory). See this example for an accessible account: https://www.react.uni-saarland.de/people/hofmann/jhofBA.pdf