The functor $g^{-1}$ is exact, hence (1)(a), (2)(a), (1)(b), and (2)(b) follow from Cohomology of Stacks, Lemmas \ref{stacks-cohomology-lemma-quasi-coherent} and \ref{stacks-cohomology-lemma-parasitic-in-terms-flat-fppf}. \medskip\noindent Proof of (1)(c) and (2)(c). The construction of $Lg_!$ in Lemma \ref{lemma-shriek-derived} (via Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-existence-derived-lower-shriek} which in turn uses Derived Categories, Proposition \ref{derived-proposition-left-derived-exists}) shows that $Lg_!$ on any object $\mathcal{H}^\bullet$ of $D(\mathcal{O}_{\mathcal{X}_{lisse,\etale}})$ is computed as $$ Lg_!\mathcal{H}^\bullet = \colim g_!\mathcal{K}_n^\bullet = g_! \colim \mathcal{K}_n^\bullet $$ (termwise colimits) where the quasi-isomorphism $\colim \mathcal{K}_n^\bullet \to \mathcal{H}^\bullet$ induces quasi-isomorphisms $\mathcal{K}_n^\bullet \to \tau_{\leq n} \mathcal{H}^\bullet$. Since $\mathcal{M}_\mathcal{X} \subset \textit{Mod}(\mathcal{X}_\etale, \mathcal{O}_\mathcal{X})$ (resp.\ $\mathcal{M}_\mathcal{X} \subset \textit{Mod}(\mathcal{O}_\mathcal{X})$) is preserved under colimits we see that it suffices to prove (c) on bounded above complexes $\mathcal{H}^\bullet$ in $D_\QCoh(\mathcal{O}_{\mathcal{X}_{lisse,\etale}})$ (resp.\ $D_\QCoh(\mathcal{O}_{\mathcal{X}_{flat,fppf}})$). In this case to show that $H^n(Lg_!\mathcal{H}^\bullet)$ is in $\mathcal{M}_\mathcal{X}$ we can argue by induction on the integer $m$ such that $\mathcal{H}^i = 0$ for $i > m$. If $m < n$, then $H^n(Lg_!\mathcal{H}^\bullet) = 0$ and the result holds. In general consider the distinguished triangle $$ \tau_{\leq m - 1}\mathcal{H}^\bullet \to \mathcal{H}^\bullet \to H^m(\mathcal{H}^\bullet)[-m] \to \ldots $$ (Derived Categories, Remark \ref{derived-remark-truncation-distinguished-triangle}) and apply the functor $Lg_!$. Since $\mathcal{M}_\mathcal{X}$ is a weak Serre subcategory of the module category it suffices to prove (c) for two out of three. We have the result for $Lg_!\tau_{\leq m - 1}\mathcal{H}^\bullet$ by induction and we have the result for $Lg_!H^m(\mathcal{H}^\bullet)[-m]$ by Lemma \ref{lemma-higher-shriek-quasi-coherent}. Whence (c) holds. \medskip\noindent Let us prove (2)(d). By (2)(a) and (2)(b) the functor $g^{-1} = g^*$ induces a functor $$ c : D_\QCoh(\mathcal{O}_\mathcal{X}) \longrightarrow D_\QCoh(\mathcal{O}_{\mathcal{X}_{flat, fppf}}) $$ see Derived Categories, Lemma \ref{derived-lemma-universal-property-quotient}. Thus we have the following diagram of triangulated categories $$ \xymatrix{ D_{\mathcal{M}_\mathcal{X}}(\mathcal{O}_\mathcal{X}) \ar[rd]^{g^{-1}} \ar[rr]_q & & D_\QCoh(\mathcal{O}_\mathcal{X}) \ar[ld]^c \\ & D_\QCoh(\mathcal{O}_{\mathcal{X}_{flat, fppf}}) \ar@<1ex>[lu]^{Lg_!} } $$ where $q$ is the quotient functor, the inner triangle is commutative, and $g^{-1}Lg_! = \text{id}$. For any object of $E$ of $D_{\mathcal{M}_\mathcal{X}}(\mathcal{O}_\mathcal{X})$ the map $a : Lg_!g^{-1}E \to E$ maps to a quasi-isomorphism in $D(\mathcal{O}_{\mathcal{X}_{flat, fppf}})$. Hence the cone on $a$ maps to zero under $g^{-1}$ and by (2)(b) we see that $q(a)$ is an isomorphism. Thus $q \circ Lg_!$ is a quasi-inverse to $c$. \medskip\noindent In the case of the lisse-\'etale site exactly the same argument as above proves that $$ D_{\mathcal{M}_\mathcal{X}}( \mathcal{X}_\etale, \mathcal{O}_\mathcal{X}) / D_{\mathcal{P}_\mathcal{X}}( \mathcal{X}_\etale, \mathcal{O}_\mathcal{X}) $$ is equivalent to $D_\QCoh(\mathcal{O}_{\mathcal{X}_{lisse,\etale}})$. Applying the last equivalence of Lemma \ref{lemma-compare-etale-fppf-QCoh} finishes the proof.