(1). Let $p$ be a prime number. Let $B\mathbb{Z}_p$ be the classifying space of the discrete group $\mathbb{Z}_p$. How to obtain $$ H^*(B\mathbb{Z}_p;\mathbb{Z}_p)=\mathbb{Z}_p[t]\otimes \Lambda[e]? $$ Here $\deg e=1$, $\deg t=2$?
(2). Let $q$ be another prime number larger than $p$. What is $$ H^*(B\mathbb{Z}_p;\mathbb{Z}_q)? $$
(3). What is $$ H^*(B\mathbb{Z}_p;\mathbb{Q})? $$
On
To simplify notation, write $\hat H^*$ for the Tate cohomology group $\hat H^*\left(\mathbb{Z}_p, \mathbb{Z}_p\right)$. Then $\hat H^*$ has period $2$; that is, the map $x\to x\cup u$ is an isomorphism $\hat H^* \to \hat H^{*+2}$ for some $u\in \hat H^2$. To prove this, consider the periodic complete resolution of $G$ via its action on $S^1$, for example, or compute $\hat H^2 = \mathbb{Z}_p$ directly and invoke duality. The low-dimensional $\hat H^*$ are easy to compute directly and then give you the required result for part (1). Parts (2) and (3) follow immediately, using the fact (proved via dimension-shifting, for example) that $\hat H^*$ vanishes if $|G| = p$ is invertible in $M$.
If you don't like using Tate cohomology, consider the Cartan-Leray spectral sequence corresponding to the fibration $S^1 \to S^1/\mathbb{Z}_p$, with $\mathbb{Z}_p$ acting by rotations. Since $H_*(S^1, \mathbb{Z}_p)$ vanishes above dimension $1$, the sequence should abut quickly. For parts (2) and (3), use the transfer map to prove that $H^n(G, M) = 0$ for $n > 0$ if $|G|$ is (finite and) invertible in $M$.
In this answer I'll assume that $\mathbb{Z}_p$ means $\mathbb{Z}/p\mathbb{Z}$.
For 2), more generally, let $G$ be a finite group and let $q$ be a prime not dividing $|G|$. Then $H^n(BG, \mathbb{F}_q)$ vanishes (for $n \ge 1$). One of many ways to see this is that under the given hypotheses $\mathbb{F}_q[G]$ is semisimple and so the category of $\mathbb{F}_q[G]$-modules has no nontrivial higher Ext groups.