Find a set of minimal natural axioms, from which we construct $\mathbb{Z}$. I am interested in this question for teaching two very different kind of students. 
The first (less important to me) is students in their first year in the university. I wish to construct $\mathbb{Z}$ by assuming as less as possible.
The second are children age $6-8$. I rememer that my teacher did it very badly and I am trying to find a way to teach children about adding and multipication in the most natural way.
Some of my thoughts: Clearly you need to assume there is $0,1$. Then by saying anumber $k$ you just mean to add $1$ to itself $k$ times.
Commutative of adding is natural to understand however I think associativity is a little bit harder. The same goes for multipication. And I belive we must take distirbutive as a axiom. 
To make a long story short I am looking for a set of minimal natural axioms, from which we construct $\mathbb{Z}$ both to students and for children.
Also, I would like to ask what are the differences between the approches you recommend.
 A: My answer only applies for teaching integers to 6-8 old children: I do not think it's a good idea to use an axiomatic approach for teaching integers to pupils. First they need to know what axioms are and how they work. Some understandings about logic is also necessary. Such an approach is in my opinion too formal and too abstract for an 7-year old child.
In the 1960s there was already a movement in mathematical education, called New Math which tried to teach mathematics in a more abstract manner. In the end this movement did not succeed. As Morris Kline wrote in his book "Why Johnny Can't Add: the Failure of the New Math":

"abstraction is not the first stage but the last stage in a mathematical development" (page.98)

For an 7-year old child I would first try to teach the intuition about integers. Let's assume the children already have an intuitive understanding about natural numbers. Now you can teach that natural numbers can be found on the number line:

File: https://commons.wikimedia.org/wiki/File:Zahlenstrahl2.gif by User:Z1 published under CC-BY 3.0
After this you can introduce integers as numbers to be found on the left side of the origin. An integer -3 means for example "going 3 steps from the origin of the left". For children this is understandable and gives them a good intuition for thinking about integers. Its also natural to name the points of the left side of the origin. A pupil might ask you about the left side of the number line beforehand. But I doubt you will find any child wondering about the axioms of integers...

File Number-line.svg by User:Hakunamenta licensed under CC0 1.0
After introducing addition $x+y$ on the number line as

Starting by $x$ and going $y$ steps to the right (when $y$ is positive) or $y$ steps to the left (if $y$ is negative) or doing nothing (if $y$ is zero)

the mentioned axioms by you like the commutative or the associative law of addition are intuitively easily to understand.
Important note: The above suggestions are only my thoughts about the topic. I am not an educationalist or I'm not educated in didactics. So I recommend you to ask your question on a forum about didactics of mathematics.
