The right to have sex - at what age?

I think you’re speaking in ignorance here. Aristotle, for example, never tried to prove any axiom. And how does Hoppe try to prove any axiom for that matter?

Proof refers to logical deduction. By definition one can’t prove an axiom (i.e., via logical deduction). Aristotel knew that. I think Hoppe knows that too.

This is all mistaken and confused.

The NAP is not an axoim, btw; it is a conclusion derived from a long chain of reasoning. It simply can’t be a philosophically satisfactory starting point.

This is mistaken in a number of ways. And I think it’s based on mistaken assumptions about metaphysics, epistemology, and ethical theory. Yes, any argument in favor of the NAP will lead back to discussions of metaphysics and epistemology because the NAP is not an axiom. And no, the NAP will not be at the base of all premises.

Your approach may be good for persuading people, good rhetorically, but it doesn’t suffice for providing a philosophical defense of the NAP.