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 $$H^{2,1}(S;\mathbb{Z}) = \mathrm{Pic}(S),$$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.

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.

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 $S$. Also, we have:$$M\mathbb{Z}^{2,1}M\mathbb{Z} = \mathrm{Pic}(S),$$the Picard group of the base scheme $S$. See Lemmas 4.8 and 6.14 in this preprint and the references therein.

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

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 $$H^{2,1}(S;\mathbb{Z}) = \mathrm{Pic}(S),$$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.