Proof of (1). Let $\mathcal{G} \subset \mathcal{F}$ with $\mathcal{F}$ a constructible sheaf of sets on $X_\etale$. Let $\eta \in X$ be a generic point of an irreducible component of $X$. By Noetherian induction it suffices to find an open neighbourhood $U$ of $\eta$ such that $\mathcal{G}|_U$ is locally constant. To do this we may replace $X$ by an \'etale neighbourhood of $\eta$. Hence we may assume $\mathcal{F}$ is constant and $X$ is irreducible. \medskip\noindent Say $\mathcal{F} = \underline{S}$ for some finite set $S$. Then $S' = \mathcal{G}_{\overline{\eta}} \subset S$ say $S' = \{s_1, \ldots, s_t\}$. Pick an \'etale neighbourhood $(U, \overline{u})$ of $\overline{\eta}$ and sections $\sigma_1, \ldots, \sigma_t \in \mathcal{G}(U)$ which map to $s_i$ in $\mathcal{G}_{\overline{\eta}} \subset S$. Since $\sigma_i$ maps to an element $s_i \in S' \subset S = \Gamma(X, \mathcal{F})$ we see that the two pullbacks of $\sigma_i$ to $U \times_X U$ are the same as sections of $\mathcal{G}$. By the sheaf condition for $\mathcal{G}$ we find that $\sigma_i$ comes from a section of $\mathcal{G}$ over the open $\Im(U \to X)$ of $X$. Shrinking $X$ we may assume $\underline{S'} \subset \mathcal{G} \subset \underline{S}$. Then we see that $\underline{S'} = \mathcal{G}$ by Lemma \ref{lemma-irreducible-subsheaf-constant-zero}. \medskip\noindent Let $\mathcal{F} \to \mathcal{Q}$ be a surjection with $\mathcal{F}$ a constructible sheaf of sets on $X_\etale$. Then set $\mathcal{G} = \mathcal{F} \times_\mathcal{Q} \mathcal{F}$. By the first part of the proof we see that $\mathcal{G}$ is constructible as a subsheaf of $\mathcal{F} \times \mathcal{F}$. This in turn implies that $\mathcal{Q}$ is constructible, see Lemma \ref{lemma-constructible-abelian}. \medskip\noindent Proof of (3). we already know that constructible sheaves of modules form a weak Serre subcategory, see Lemma \ref{lemma-constructible-abelian}. Thus it suffices to show the statement on submodules. \medskip\noindent Let $\mathcal{G} \subset \mathcal{F}$ be a submodule of a constructible sheaf of $\Lambda$-modules on $X_\etale$. Let $\eta \in X$ be a generic point of an irreducible component of $X$. By Noetherian induction it suffices to find an open neighbourhood $U$ of $\eta$ such that $\mathcal{G}|_U$ is locally constant. To do this we may replace $X$ by an \'etale neighbourhood of $\eta$. Hence we may assume $\mathcal{F}$ is constant and $X$ is irreducible. \medskip\noindent Say $\mathcal{F} = \underline{M}$ for some finite $\Lambda$-module $M$. Then $M' = \mathcal{G}_{\overline{\eta}} \subset M$. Pick finitely many elements $s_1, \ldots, s_t$ generating $M'$ as a $\Lambda$-module. (This is possible as $\Lambda$ is Noetherian and $M$ is finite.) Pick an \'etale neighbourhood $(U, \overline{u})$ of $\overline{\eta}$ and sections $\sigma_1, \ldots, \sigma_t \in \mathcal{G}(U)$ which map to $s_i$ in $\mathcal{G}_{\overline{\eta}} \subset M$. Since $\sigma_i$ maps to an element $s_i \in M' \subset M = \Gamma(X, \mathcal{F})$ we see that the two pullbacks of $\sigma_i$ to $U \times_X U$ are the same as sections of $\mathcal{G}$. By the sheaf condition for $\mathcal{G}$ we find that $\sigma_i$ comes from a section of $\mathcal{G}$ over the open $\Im(U \to X)$ of $X$. Shrinking $X$ we may assume $\underline{M'} \subset \mathcal{G} \subset \underline{M}$. Then we see that $\underline{M'} = \mathcal{G}$ by Lemma \ref{lemma-irreducible-subsheaf-constant-zero}. \medskip\noindent Proof of (2). This follows in the usual manner from (3). Details omitted.