I am trying to understand spectra and their relation to cohomology theory. I have read that there is an essentially surjective functor $\mathrm{coTheo} \to \mathrm{Spec}$ from the category of cohomology theories into the category of spectra, by sending each cohomology theory to its representing sequence of spaces, which exist by Browns representability theorem.
On the other hand, when we have a spectrum $E$, then we define $E^n(Z) := [\Sigma^{\infty} Z, \Sigma^n E]$. I have a few questions regarding this.
1)
I have identified $[\Sigma^{\infty} Z, \Sigma^nE]$ with $[Z, E^n]$ for the essential surjectivity (since a morphism from $\Sigma^{\infty} Z$ is given by the morphism from $Z$ itself), but why not just define $E^n(Z)$ as the latter?
2)
Are the sets $[\Sigma^{\infty} Z, \Sigma^n E]$ always abelian groups?
And further, is my above identification between cohomology theories and representing spectra true?