The set of Locales is a subset of the set of Toposes. Given an ordered set $(Z, \leq)$, we can consider the category $C_Z$ with $Ob(C)=Z$ and for each $r,e\in Z$, $\hom_{C_Z}(r,e)$ is a singleton if $r\leq e$ and is $\emptyset$ if $\neg(r\leq e)$. The idea is that ordered sets are special kinds of categories (with no more than one morphism between any two objects).
An ordered set is called a frame if it admits arbitrary joins and binary meets and binary meets distribute over joins. A morphisms between frames is a monotone function preserving arbitrary joins and binary meets. A Grothendieck topos is a category satisfying the so-called Giraud axioms, which are very similar to the axioms of a frame (binary limits; arbitary small colimits, which are stable under pullbacks).

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*Is $Z\mapsto C_Z$ a fully faithful functor from the category of frames to the category of Grothendieck toposes (with geometric morphisms)?


*If yes, why is it more common to embed frames into Grothendieck toposes via $Z\mapsto Sh(Z)=$ topos of sheaves on $Z$? Is this embedding somehow related to the functor $Z\mapsto C_Z$?


*Does 1. imply that the category of frames is equivalent to the category of Grothendieck toposes of the form $C_Z$ for some ordered set?


*Why are frames required to be ordered sets, i.e., such that $\leq$ is antisymmetric: if $r\leq e\leq r$, then $r=e$ and not merely preordered sets? Because for Grothendieck toposes we don't require them that isomorphic objects are equal (which is the categorical analogue).

No, for a frame $Z$, the category $C_Z$ is never a Grothendieck topos (unless it has only one object).
In terms of Giraud's axioms, $C_Z$ fails the axiom that coproducts should be disjoint. In a frame, $1+1=1$, while in a topos, $1+1=1$ implies $0=1$, which implies that the topos is trivial.
