Proof.
Proof of (1). Since $\text{cd}(f) < \infty $ we see that
\[ f_* : \textit{Mod}(X_{\acute{e}tale}, \mathbf{Z}/n\mathbf{Z}) \longrightarrow \textit{Mod}(Y_{\acute{e}tale}, \mathbf{Z}/n\mathbf{Z}) \]
has finite cohomological dimension in the sense of Derived Categories, Lemma 13.32.2. Let $I$ be a set and for $i \in I$ let $E_ i$ be an object of $D(X_{\acute{e}tale}, \mathbf{Z}/n\mathbf{Z})$. Choose a K-injective complex $\mathcal{I}_ i^\bullet $ of $\mathbf{Z}/n\mathbf{Z}$-modules each of whose terms $\mathcal{I}_ i^ n$ is an injective sheaf of $\mathbf{Z}/n\mathbf{Z}$-modules representing $E_ i$. See Injectives, Theorem 19.12.6. Then $\bigoplus E_ i$ is represented by the complex $\bigoplus \mathcal{I}_ i^\bullet $ (termwise direct sum), see Injectives, Lemma 19.13.4. By Lemma 59.51.7 we have
\[ R^ qf_*(\bigoplus \mathcal{I}_ i^ n) = \bigoplus R^ qf_*(\mathcal{I}_ i^ n) = 0 \]
for $q > 0$ and any $n$. Hence we conclude by Derived Categories, Lemma 13.32.2 that we may compute $Rf_*(\bigoplus E_ i)$ by the complex
\[ f_*(\bigoplus \mathcal{I}_ i^\bullet ) = \bigoplus f_*(\mathcal{I}_ i^\bullet ) \]
(equality again by Lemma 59.51.7) which represents $\bigoplus Rf_*E_ i$ by the already used Injectives, Lemma 19.13.4.
Proof of (2). This is identical to the proof of (1) and we omit it.
$\square$
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