Start with the second question:
Why is the first-order version of Peano's axioms better to describe the true nature of natural numbers?
Careful! First order PA [PA1] is, as said, not categorical, and has non-standard models (models which are not isomorphic to the 'true' natural numbers, because -- as well as having a 'zero' and its 'successors' -- they have additional, non-standard, elements that come 'after' the zero and its successors). Since first-order PA can't distinguish its minimal model which does just have a zero and its successors from the models with unintended surplus content, it doesn't pin down "the true nature of natural numbers", and can't be sold as doing so.
So, it might naturally be asked, why do we at least start doing formal arithmetic by investigating PA1 rather than looking at some arithmetic PA2 with a second-order logic? More generally,
Why is it better to work with first-order logic than with second-order logic?
But remember: second-order semantic consequence [with "full semantics"] can't be recursively axiomatized -- i.e. there is no nice formal deductive system where we can effectively decide what is an axiom, and what is a well-constructed proof etc., such that a conclusion can be formally deduced from some axioms in the proof system just if the conclusion semantically follows from the axioms. Which means that any effectively axiomatised formal deductive system of second-order arithmetic capture all the semantic consequences of the second-order Peano axioms. It will in fact be negation incomplete. So in one good sense it won't be a complete theory of the nature of the numbers either.
True, an axiomatized second order arithmetic will be stronger than PA1, will prove more, so why not go straight to discuss such a stronger theory (it will tell us more about the numbers even if it doesn't fully pin down their "true nature")? Well, for natural pedagogic reasons, we'll want to start with the simpler theory! So it is entirely natural to start our investigations of arithmetic by (1) investigating PA1 and (2) showing by contrast that second-order arithmetic with full semantics is not axiomatizable [two standard ingredients of a first course] before (3) looking at axiomatizable versions of PA2 of various strengths which extend PA1.