First of all, if one is describing the integers axiomatically, the axiom is not that "$a\cdot 1=a$ for all $a$", but rather that there exists a number "$1$" that has that property (one can then proceed to prove that it must be the only number with that property).
Similarly, the corresponding axiom for addition is that there exists some number "$0$" with the property that $a+0=a$ for any $a$.
However, looking in the book, I can see that Spivak is listing properties of the integers, some phrased as axioms, but he is taking the integers as already having been constructed, so that he doesn't need to say things in the way I did above. For example, he says
At any rate, to answer your first question, by the axiom that multiplication distributes over addition, we know that for any $a$ we have
$$a\cdot (0+0)=a\cdot 0+a\cdot 0$$
But $0+0=0$ because $0$ (by definition) is an additive identity, so
$$a\cdot 0=a\cdot 0+a\cdot 0$$
Whatever $a\cdot 0$ is, we know it has an additive inverse, which we can now add to both sides:
$$(a\cdot 0)+(-(a\cdot 0))=(a\cdot 0)+(a\cdot 0)+(-(a\cdot 0))$$
Also, the statement that $a=b$ means that they are identical, that there is a single number that we have given two different names to. Any expression whatsoever involving $a$ can have $b$ substitued for $a$ in it, and vice versa. That is simply what equality means - there is no unstated axiom.