I can only answer some of your questions.

Yes, the Zariski locale is extensively studied. It's one of the ways of setting up scheme theory in a constructive context: Don't define schemes as locally ringed spaces, but as locally ringed locales. The locally ringed locale $\mathrm{Spec}(A)$ always enjoys the universal property we expect of it, namely that morphisms $X \to \mathrm{Spec}(A)$ of locally ringed locales are in one-to-one correspondence with ring homomorphisms $A \to \mathcal{O}_X(X)$. (Only) if the Boolean Prime Ideal Theorem is available (a slightly weaker form of the axiom of choice), one can show that the Zariski locale has *enough points*. In this case it's isomorphic to the locale induced from the classical topological space of prime ideals (equipped with the Zariski topology).

(Note that the preceding paragraph assumes that you define the Zariski locale of a ring $A$ to be the locale of *prime filters* of $A$, not the locale of *prime ideals*. (A prime filter is a direct axiomatization of what's classically simply the complement of a prime ideal.) The locale of prime ideals also exists, but does not coincide with the true Zariski locale; classically, it is isomorphic to the topological space of prime ideals equipped with the flat topology.)

Yes, there is a general theory of locales as spaces of imaginary points. Briefly, to any *propositional geometric theory* $T$ (roughly speaking, a collection of axioms of a certain form, such as the axioms of a prime ideal or of a prime filter), there is a *classifing locale* $\mathrm{Set}[T]$. The points of this locale are exactly the $\mathrm{Set}$-based models of $T$ (that is, the actual prime ideals or the actual prime filters). It can happen that there are no such models, yet still the theory $T$ is consistent. In this case the classifying local doesn't have any points, yet still is not the trivial locale.

Any locale $X$ is the classifying locale of some propositional geometric theory, a theory which deserves the name "theory of points of $X$".

The theory of classifying locales indeed allows you to construct spaces (locales) of things which aren't expected to exist, just by writing down the axioms of the hypothetical objects. A particularly tantalizing example is the locale of surjections from $\mathbb{N}$ to $\mathbb{R}$. There are no such surjections, of course, therefore this locale doesn't have any points. But it's still nontrivial. The topos of sheaves over this locale contains an epimorphism from $\underline{\mathbb{N}}$, the constant sheaf with stalks $\mathbb{N}$, to $\underline{\mathbb{R}}$; this epimorphism could be named the "walking surjection from $\mathbb{N}$ to $\mathbb{R}$", as any such surjection in any topos is a pullback of this one.

(The reals starred in the preceding paragraph only for dramatic purposes. The previous paragraph stays correct if $\mathbb{R}$ is substituted by any inhabited set. The described construction has been used, as one of a series of reduction steps, in Joyal and Tierney's celebrated monograph *An Extension of the Galois Theory of Grothendieck*.)

An excellent entry point to the business of locales as spaces of ideal points is [a very accessible expository note](https://www.cs.bham.ac.uk/~sjv/LocTopSpaces.pdf) by Steve Vickers. (When you've finished with this one, be sure to check out his further surveys, all available on his webpage, for instance [this one](https://www.cs.bham.ac.uk/~sjv/GeoAspects.pdf).)