Logical "axioms" are usually not "selected", they codify forms of inference prevailing in a practice with some tweaks for uniformity, applicability, etc., purposes. For example, classical logic revises the natural language conditional to make it truth functional, i.e. dependent only on the truth values of the terms. This makes for a handier calculus, and is more appropriate in mathematics. One can play around with other tweaks, and bigger revisions.
The issue of justifying logic is subtle. If it is in the traditional sense of justification, one would presumably need to use logic in the process of justification, which would make the enterprise circular or lead to infinite regress. Intuition can not provide a backstop because having an intuitive feeling of certainty is no justification of certainty, and people are at times unwaveringly certain about all sorts of dubious things. The justification can not be empirical either, for then logic would offer no more assurance than inductive generalization. And we clearly treat logic differently, as a norm, not subject to refutation.
One can compare different logical systems according to some pragmatic criteria and select ones that better suit some purpose at hand. Classical logic does pretty well for most ordinary purposes, and it has the advantage of simplicity. Harmony is a convenient technical property to have, it favors intuitionistic logic. Ensuring successful termination of disputes is yet another virtue, at the core of Girard's justificational program of "transcendental syntax", see What are the current topics in philosophy of logic? It favors something like linear logic. Of course, none of these is an "ultimate justification", selection of the criterion can be questioned just as the logic it favors. The regress still looms.
The root cause of the regress is that traditional kind of justification always transcribes words into more words, sentences are justified by more sentences, propositions by more propositions. One needs to step out of this neverending linguistic sea. Thereby, beyond the boundary, lies the ultimate "justification" broadly construed, one that underlies logic, intuition, and rationality itself, as well as any norms we adopt. The inescapable pressure of reality. Here is Peirce's description of what he calls the "English doctrine" as opposed to the "German doctrine" of intuitive justification:
"But in truth the essence of the matter lies in a nutshell. Facts are hard things which do not consist in my thinking so and so, but stand unmoved by whatever you or I or any man
or generations of men may opine about them. It is those facts that I want to know, so that I may avoid disappointments and disasters. Since they are bound to press upon me at last, let me know them as soon as possible, and prepare for them. This is, in the last analysis, my whole motive in reasoning. Plainly, then, I wish to reason in such way that the facts shall not, and cannot, disappoint the promises of my reasoning.
Whether such reasoning is agreeable to my intellectual impulses is a matter of no sort of consequence. I do reason not for the sake of my delight in reasoning, but solely to avoid disappointment and surprise. Consequently, I ought to plan out my reasoning so that I evidently shall avoid those surprises. That is the rationale of the English
doctrine. It is as perfect as it is simple.
[...]I believe that our natural judgments as to what is reasonable are due to thinking over, ordinarily in a more or less confused way, what would happen. We imagine cases, place mental diagrams before our mind's eye, and multiply these cases, until a habit is formed of expecting that always to turn out the case, which has been seen to be the result in all the diagrams. To appeal to such a habit is a very different thing from appealing to any immediate instinct of rationality... If it be so, the German doctrine falls to the ground; for to form a notion of right reasoning from diagrams showing what will happen, is to form that notion virtually according to the English doctrine of logic, by reasoning from the nature of things."
