$(i) \Leftrightarrow (ii)$ is true and is Proposition C.2.4.14 in Peter Johnstone's Sketches of an elephant.
$(ii)$ and $(iii)$ are not equivalent: there is a large gap between "the diagona is a mono" and "the diagonal is an embeddings"
For a typical example, take a free but non-proper action of a group $G$ on a locale (or space) $X$. To fix the idea, take $G = \mathbb{Z}$ acting on $X=S^1$ the unit circle by rotation by an irrational angle.
The topos of equivariant sheaves $X//G$ classifies "orbits of $G$", i.e. a $G$-torsor $T$ (a principale $G$-bundle) together with a $G$-equivariant map $T \to G$. Because the action is free, the category of point in any topos will have no non-trivial morphisms.
But that topos is not localic at all: its subterminal objects are the $G$-invariant open subset so in our concrete example its only $\emptyset$ and $1$.
One can also compute the diagonal map. $\mathcal{T} \times \mathcal{T}$ can be shown to be the topos corresponding to the action of $G \times G$ on $X \times X$. Subtopos of this would corresponds to $G \times G$-equivariant sublocales of $X \times X$ and the diagonal is not $G \times G$-equivariant.
Of course some of the claim I made above would require a proof... but that might be a bit too long for MO.