For the topological question, [this thread][1] and [this thread][2] have interesting answers and references about integral (co)homology of Eilenberg-MacLane spaces and spectra. Let me add a few comments.

I'll focus on integral *homology*. Let $A$ be an abelian group and consider the stable homology group $$H\mathbb{Z}_n HA = \pi_n(H\mathbb{Z} \wedge HA) \cong H_{m+n}(K(A,m);\mathbb{Z})$$ for $m > n$ (the stable range). [Eilenberg and MacLane][3] give us: $$H\mathbb{Z}_n HA = \begin{cases}
A &n = 0 \\
0 &n = 1 \\
A/2 &n = 2 \\
\mbox{}_2 A &n = 3 \\
A/2 \oplus A/3 &n = 4 \\
\mbox{}_2 A \oplus \mbox{}_3 A &n = 5 \\
\end{cases}$$
respectively in Theorems 20.3, 20.5, 23.1, 24.1, 25.1, and 25.3. (Ok, the cases $n=0,1$ are rather an exercise.) Here, $\mbox{}_2 A$ denotes the $2$-torsion subgroup of $A$. Sections 26-27 address the cohomology $H^*(K(A,m); \mathbb{Z})$ for small $m$.

Section 5 of [Cartan][4] and Section 6 of this [Séminaire Henri Cartan][5] provide a way to compute $H\mathbb{Z}_n HA$ entirely. It is a torsion group whose $p$-primary part is a finite direct sum of copies of $A/p$ and $\mbox{}_p A$ indexed by some combinatorial formulas.

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For the motivic question, I'll defer to the algebraic geometers. Here are a few ideas.

For a nice base scheme $S$ (say, essentially smooth over a field), we have$$M\mathbb{Z}^{0,0}M\mathbb{Z} = \mathbb{Z}^{\pi_0(S)},$$where $\pi_0(S)$ denotes the set of connected components of the base scheme $S$. See Lemma 6.14 in [this preprint][6] and the references therein.

In Lemma 4.10, we had an $M\mathbb{Z}$-*module* map $M\mathbb{Z} \to \Sigma^{2,1} M\mathbb{Z}$, which corresponds to a map $\mathbf{1} \to \Sigma^{2,1} M\mathbb{Z}$ in $SH(S)$. Those are given by $$H^{2,1}(S;\mathbb{Z}) = \mathrm{Pic}(S),$$the Picard group of the base scheme $S$ (see Lemma 4.8). I'm not sure what maps $[M\mathbb{Z}, \Sigma^{2,1} M\mathbb{Z}]$ in $SH(S)$ look like.

These [lecture notes][7] by Spitzweck might be helpful, especially Section 3. [This book][8] by Mazza, Voevodsky, and [Weibel][9] is also a nice reference.


  [1]: https://mathoverflow.net/questions/24754/cohomology-of-the-eilenberg-maclane-spaces-kg-n
  [2]: https://mathoverflow.net/questions/50519/integral-cohomology-stable-operations
  [3]: https://doi.org/10.2307/1969702
  [4]: https://doi.org/10.1073/pnas.40.8.704
  [5]: http://archive.numdam.org/item/SHC_1954-1955__7_1_A11_0/
  [6]: https://arxiv.org/pdf/1711.05230v3.pdf
  [7]: https://www.him.uni-bonn.de/fileadmin/him/Spitzweck-minicourse-HIM.pdf
  [8]: https://www.claymath.org/publications/monographs/lecture-notes-motivic-cohomology
  [9]: http://sites.math.rutgers.edu/~weibel/motiviclectures.html