For the topological question, this thread and this thread 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 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 and Section 6 of this Séminaire Henri Cartan 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.

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 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 $$[\mathbf{1}, \Sigma^{2,1} M\mathbb{Z}] = H^{2,1}(S;\mathbb{Z}) = \mathrm{Pic}(S),$$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 by Spitzweck might be helpful, especially Section 3. This book by Mazza, Voevodsky, and Weibel is also a nice reference.