Axioms necessary for theoretical computer science
Yes, the topic has been studied in proof complexity. It is called Bounded Reverse Mathematics. You can find a table containing some reverse mathematics results on page 8 of Cook and Nguyen's book, "Logical Foundations of Proof Complexity",  2010. Some of Steve Cook's previous students have worked on similar topics, e.g. Nguyen's thesis, "Bounded Reverse Mathematics", University of Toronto, 2008.

Alexander Razborov (also other proof complexity theorists) has some results on the weak theories needed to formalize the circuit complexity techniques and prove circuit complexity lowerbounds. He obtains some unprovability results for weak theories, but the theories are considered too weak.

All of these results are provable in $RCA_0$ (Simpson's base theory for Reverse Mathematics), so AFAIK we don't have independence results from strong theories (and in fact such independence results would have strong consequences as Neel has mentioned, see Ben-David's work (and related results) on independence of $\mathbf{P} vs. \mathbf{NP}$ from $PA_1$ where $PA_1$ is an extension of $PA$).