It's probably a silly question but I’ve been carrying this one since infancy so i might as well ask it already.
let ($p \implies q$) be a theorem where $p$ is the hypotheses and $q$ is the conclusion. If stated in logical symbols can it always be reduced to the axioms by logical manipulation?
Does that mean that if different rules of logic are chosen (say quantum logic) one cannot prove theorems the "normal" (language and common sense) way since intuition fails?
This question is quite natural, as the notion of "rigorous proof" largely depends on the context in which it is mentioned.
The short answer is Yes. Every mathematical proof, if correct, can be formulated as a derivation starting from axioms (usually of ZFC), and using basic deduction rules.
In practice, it is of course infeasible, so what we call a "proof" is a description containing enough information to convey the ideas necessary to build this rigorous proof. This is (almost) never done, but should be always possible. Therefore, what is sufficient to constitute a "proof" highly depends on the interlocutors exchanging it, because they have to be able to fill the gaps with their knowledge. For instance, a proof of most highschool problems is reduced to "trivial" for professional mathematicians.
In the end, the answer is "yes", but you are wrong with your last sentence: a "human" proof is transformed into a "logical" one not by logical manipulation, but by filling in a lot of gaps, that are omitted in human formulations.
If, as you mention, different rules for logic are chosen, the proofs need to be compatible with these new rules, i.e. extendable into logical proofs following these rules.