Proof of (1). The equality $H^n_{Zar}(X, \pi_X^{-1}\mathcal{F}) = H^n(X, \mathcal{F})$ is a general fact coming from the trivial observation that coverings of $X$ in $\textit{LC}_{Zar}$ are the same thing as open coverings of $X$. The reader who wishes to see a detailed proof should apply Lemma \ref{lemma-cohomology-bigger-site} to the functor $X_{Zar} \to \textit{LC}_{Zar}$. \medskip\noindent Proof of (2). This is true because $\pi_{X, *} = \tau_X^{-1}$ for some morphism of topoi $\tau_X : \Sh(X_{Zar}) \to \Sh(\textit{LC}_{Zar})$ as follows from Sites, Lemma \ref{sites-lemma-bigger-site} applied to the functor $X_{Zar} \to \textit{LC}_{Zar}/X$ used to define $\pi_X$. \medskip\noindent Proof of (3). This is true because $\tau_X^{-1} \circ \pi_X^{-1}$ is the identity functor by Sites, Lemma \ref{sites-lemma-bigger-site}. Or you can deduce it from the explicit description of $\pi_X^{-1}$ in Lemma \ref{lemma-describe-pullback-pi}. \medskip\noindent Proof of (4). Apply (3) to an complex of abelian sheaves representing $K$. \medskip\noindent Proof of (5). The morphism of topoi $f_{Zar}$ comes from an application of Sites, Lemma \ref{sites-lemma-relocalize} and in our case comes from the continuous functor $Z/Y \mapsto Z \times_Y X/X$ by Sites, Lemma \ref{sites-lemma-relocalize-given-fibre-products}. The diagram commutes simply because the corresponding continuous functors compose correctly (see Sites, Lemma \ref{sites-lemma-composition-morphisms-sites}). \medskip\noindent Proof of (6). We have $H^n_{Zar}(X, \pi_Y^{-1}\mathcal{G}) = H^n_{Zar}(X, f_{Zar}^{-1}\pi_Y^{-1}\mathcal{G})$ for $\mathcal{G}$ in $\textit{Ab}(Y)$, see Lemma \ref{lemma-cohomology-of-open}. This is equal to $H^n_{Zar}(X, \pi_X^{-1}f^{-1}\mathcal{G})$ by the commutativity of the diagram in (5). Hence we conclude by (1) in the case $L$ consists of a single sheaf in degree $0$. The general case follows by representing $L$ by a bounded below complex of abelian sheaves. \medskip\noindent Proof of (7a). Let $\mathcal{F}$ be a sheaf on $X$. Let $g : Z \to Y$ be an object of $\textit{LC}_{Zar}/Y$. Consider the fibre product $$ \xymatrix{ Z' \ar[r]_{f'} \ar[d]_{g'} & Z \ar[d]^g \\ X \ar[r]^f & Y } $$ Then we have $$ (f_{Zar, *}\pi_X^{-1}\mathcal{F})(Z/Y) = (\pi_X^{-1}\mathcal{F})(Z'/X)  = \Gamma(Z', (g')^{-1}\mathcal{F})  = \Gamma(Z, f'_*(g')^{-1}\mathcal{F}) $$ the second equality by Lemma \ref{lemma-describe-pullback-pi}. On the other hand $$ (\pi_Y^{-1}f_*\mathcal{F})(Z/Y) = \Gamma(Z, g^{-1}f_*\mathcal{F}) $$ again by Lemma \ref{lemma-describe-pullback-pi}. Hence by proper base change for sheaves of sets (Cohomology, Lemma \ref{cohomology-lemma-proper-base-change-sheaves-of-sets}) we conclude the two sets are canonically isomorphic. The isomorphism is compatible with restriction mappings and defines an isomorphism $\pi_Y^{-1}f_*\mathcal{F} = f_{Zar, *}\pi_X^{-1}\mathcal{F}$. Thus an isomorphism of functors $\pi_Y^{-1} \circ f_* = f_{Zar, *} \circ \pi_X^{-1}$. \medskip\noindent Proof of (7b). Let $K \in D^+(X)$. By Lemma \ref{lemma-unbounded-describe-higher-direct-images} the $n$th cohomology sheaf of $Rf_{Zar, *}\pi_X^{-1}K$ is the sheaf associated to the presheaf $$ (g : Z \to Y) \longmapsto H^n_{Zar}(Z', \pi_X^{-1}K) $$ with notation as above. Observe that \begin{align*} H^n_{Zar}(Z', \pi_X^{-1}K) & = H^n(Z', (g')^{-1}K) \\ & = H^n(Z, Rf'_*(g')^{-1}K) \\ & = H^n(Z, g^{-1}Rf_*K) \\ & = H^n_{Zar}(Z, \pi_Y^{-1}Rf_*K) \end{align*} The first equality is (6) applied to $K$ and $g' : Z' \to X$. The second equality is Leray for $f' : Z' \to Z$ (Cohomology, Lemma \ref{cohomology-lemma-before-Leray}). The third equality is the proper base change theorem (Cohomology, Theorem \ref{cohomology-theorem-proper-base-change}). The fourth equality is (6) applied to $g : Z \to Y$ and $Rf_*K$. Thus $Rf_{Zar, *}\pi_X^{-1}K$ and $\pi_Y^{-1}Rf_*K$ have the same cohomology sheaves. We omit the verification that the canonical base change map $\pi_Y^{-1}Rf_*K \to Rf_{Zar, *}\pi_X^{-1}K$ induces this isomorphism.