Can we prove that an extension to a structure is consistent if the original structure is? As I understand it, there is no way to prove that $\mathbb{N}$, as modeled by P.A., is consistent - meaning it may be possible to demonstrate eg. $5 = 3$. Therefore it is presumably also impossible to prove that $\mathbb{Z}$ - the extension of $\mathbb{N}$ with additive inverses - is consistent.
But could we at least prove that $\mathbb{Z}$ is consistent if $\mathbb{N}$ is? Could we do the same for $\mathbb{Q}$, $\mathbb{R}$, $\mathbb{C}$?
 A: I think you have this backwards: structures are neither consistent or inconsistent, theories are!  Furthermore, given any structure $\mathfrak{A}$ over a (first-order) language, the set $\mathrm{Th} ( \mathfrak{A} )$ of all sentences true in $\mathfrak{A}$ is consistent.
The point of Gödel's (Second) Incompleteness Theorem is that unless PA is inconsistent, there is no way of proving the consistency of PA without transcending PA itself.  To quote George Boolos:

It can be proved that if it can be proved that it can't be proved that two plus two is five, then it can be proved that two plus two is five, and math is a lot of bunk.

Consequently, there is no way to prove that models of PA exist (meaning that there is no way of demonstating that a particular structure $\mathfrak{N}$ satisfies all of the axioms of PA) without transcending PA.
However, it is possible to jump to significantly stronger theories to prove the existence of models of PA.  As an example, ZF proves the existence of the smallest inductive set (a set $I$ such that $\emptyset \in I$ and $x \cup \{ x \} \in I$ for all $x \in I$), call it $N$, and furthermore via a natural translation from the language of PA into the language of set theory, ZF proves that $N$ satisfies all of the axioms of PA.
In such strong enough set theories, we may then unproblematically construct the integers, the rationals, etc.
This may be seen as simply "passing the buck": While a strong enough theory $T$ may prove the consistency of PA, this just means that the consistency of $T$ implies the consistency of PA.  In particular, $T$ itself will be subject to Gödel's Incompleteness Theorem (as long as it is sufficiently simply presented).
A: I think that what you intended to ask is a bit different than what you had asked. So, I'll rephrase your question first. It is the case that the existence of a model of $\mathbb N$ cannot be proved without making some assumption about a universe outside of $\mathbb N$. Put a bit differently, it is impossible to display a model for the natural numbers without making use of some more fundamental notion, which we must take on faith. 
Your question is then whether one can exhibit a model for the integerst provided one has a model for the natural numbers, and similarly for the rationals and the reals, and beyond. The answer is yes, it can be done, in many different yet essentially the same way. Moreover, there are various paths one can take, and luckily all the intermediate stops give essentially the same notions. 
So, for instance, suppose you have a model $\mathbb  N$ for the naturals satisfying Peano's axioms. You can first introduce the integers by considering the follow formal construction: Let $X=\mathbb N \times \mathbb N$, and define the equivalence relation $(x,y)\equiv(u,v)$ precisely when $x+v=u+y$. Then one can show that the quotient is a model for the integers. Or, you can first introduce the non-negative rationals by considering $Y=\{(x,y)\mid y\ne 0\}$ and the equivalence relation $(x,y)\equiv(u,v)$ precisely when $xv=yu$. Now the quotient will be a model for $\mathbb Q_+$. These are very standard constructions, and can (with a bit of care) be performed interchangeably to produce an essentially unique (that means, any two are isomorphic) model of the rationals. Now, one can complete the rationals in any of several ways to arrive at an essentially unique model of the reals. Then one can construct the complex numbers, and so on. 
So, if a model of the naturals exists then it can (within some ambient set theory) be used to construct models of the integers, rationals, reals, and so on. 
