Deduction Theorem proof issue

Question

I restudied deeply logic (from scratch) and set theory (ZFC) but I came across 2 issues:

1. In propositional logic, we prove the Deduction theorem using induction, but induction itself is proved using logical principles based on deduction, doesn't that make it a circular proof ?
2. In the development of logic we use sets, but in the development of set theory we use logic, so again is it a circular proof ?