Given a set $S$, let $Top(S)$ denote the partially ordered set (poset) of topologies on $S$, ordered by fineness, so the discrete topology, $Disc(S)$, is maximal. Given a presheaf $Q$ on $Disc(S)$, let $$Top_{Q}(S)\subseteq Top(S)$$ denote the subposet of topologies on $S$ for which $Q$ is a sheaf.

For example, if $1$ is the terminal presheaf then $Top_{1}(S)=Top(S)$, i.e., every topology makes $1$ a sheaf. In general $Top_{Q}(S)$ may be empty, e.g., if $Q$ assignes a non-terminal set $Q(\emptyset)\not\cong\{*\}$ to the emptyset $\emptyset\subseteq S$.

Question 1: In general, what can one say about the poset $Top_Q(S)$? For example, is it closed under binary meets or joins in $Top(S)$?

Let $Top^{sep}_Q(S)$ denote the poset of $S$-topologies on which $Q$ is a separated presheaf. Recall that a presheaf on a space $X$ is separated if every matching family of sections on a cover extends to at most one section on their union. This condition is less stringent than the sheaf condition, which replaces at most one with exactly one. In general, we have $$Top_Q(S)\subseteq Top^{sep}_Q(S).$$ For example the initial presheaf $0$ on $Disc(S)$ is a separated presheaf but not a sheaf, so $Top_0(S)\subsetneq Top^{sep}_0(S)$.

Question 2: What can one say about the poset $Top^{sep}_Q(S)$?