How can Computer Science theories and inquiries be resolved?
Theoretical computer science, to which the questions you state belong, is part of mathematics, and the methods used to resolve questions are the same ones used elsewhere in math, namely the method of proofs. The questions you list are notoriously difficult, and seem to be out of reach for current methods.

In the past, the methods used to attack questions of the sort you describe were purely combinatorial. Recently, Ketan Mulmuley and his students have come up with a different method which employs representation theory (via some algebraic geometry), and is conjectured by them to be one day strong enough to resolve the P versus NP question. Currently, though, the only questions within reach of their methods are algebraic ones like permanent versus determinant rather than Boolean ones like P versus NP.

Since it is widely (though not universally) believed that P is different from NP, many results are proved conditional on this assumption. That is, they hold only if P is indeed different from NP. The area of hardness of approximation, which studies how well different combinatorial problems can be approximated in the worst case (in polynomial time), is a good example: all results in this area are conditional on P versus NP or on some even stronger assumptions.