This is described in the paper
Bunge and Carboni, The symmetric topos, Journal of Pure and Applied Algebra 105:233-249, 1995.
which describes the sense in which the construction you call Spec is analogous to the symmetric algebra construction.
Bunge and Carboni give a biadjunction between the bicategory R, which is the opposite of the bicategory of Grothendieck toposes, and the bicategory A of locally presentable categories and cocontinuous functors (equivalently, left adjoints).
The following argument shows that if $C$ is essentially large (and locally small), then the presheaf $\Omega$ of sieves on $C$ is not small. Unfortunately, as Zhen points out, this does not show that $\mathcal{P}C$ does not have a subobject classifier, since its subobject classifier ought to be the presheaf of small sieves (i.e., sieves such that the corresponding presheaf is a small presheaf). In general, not all sieves are small, and in fact it seems rather difficult to construct any small sieves (exercise: construct a locally small category in which every object has a large set of sieves, but not a single nontrivial small sieve). It thus seems unlikely to me that this argument can be turned into a proof that $\mathcal{P}C$ does not have a subobject classifier without some strong additional hypotheses. Nevertheless, I'm keeping it as an answer in case you may find it somehow helpful (and because it is a nifty argument, even thought it doesn't prove what I hoped it would!).
Suppose that $\Omega$ is a small presheaf. Then every object $a\in C$ has only a small set of sieves, and there is a small set $S$ of objects such that for every $a\in C$ and every sieve $\sigma\in\Omega(a)$, there is an object $s\in S$, a map $f:a\to s$, and a sieve $\tau\in\Omega(s)$ such that $\sigma=f^*\tau$. In fact, if any such $\tau$ exists, then there is a canonical choice, namely $\tau=f_*\sigma$.
For any $a\in C$, let $\sigma_a$ be the sieve consisting of all maps $b\to a$ that do not have a right inverse (this is the second-largest sieve on $a$). We can thus find some $s_a\in S$ and $f_a:a\to s_a$ such that $\sigma_a=f_a^*f_{a*}\sigma_a$. Let's figure out explicitly what the condition $\sigma_a=f_a^*f_{a*}\sigma_a$ means. The sieve $f_{a*}\sigma_a$ consist of all maps of the form $f_ag$ where $g:b\to a$ does not have a right inverse. The sieve $f_a^*f_{a*}\sigma_a$ then consists of all maps $h$ such that $f_ah=f_ag$ for some $g$ that does not have a right inverse. The equation $\sigma_a=f_a^*f_{a*}\sigma_a$ holds iff the latter sieve does not contain the identity $1:a\to a$, or equivalently if every $g:a\to a$ such that $f_a=f_ag$ has a right inverse.
Since $S$ is small and $C$ is essentially large, we can find an essentially large set of objects $D\subseteq C$ such that all the objects $a\in D$ have the same associated object $s_a\in S$. That is, there is a single object $s\in S$ such that for each $a\in D$, there is a map $f_a:a\to s$ such that if $g:a\to a$ is such that $f_a=f_ag$, then $g$ has a right inverse. Now consider the sieves on $s$ generated by these maps $f_a$. Since $\Omega(s)$ is a small set, there is an essentially large set of objects $E\subseteq D$ such that the $f_a$ for $a\in E$ all generate the same sieve on $s$. That is, for any $a,b\in E$, there exist maps $g:a\to b$ and $h:b\to a$ such that $f_bg=f_a$ and $f_ah=f_b$. But then $f_ahg=f_a$, so $hg$ has a right inverse, and similarly $gh$ has a right inverse. In particular, $g$ and $h$ both have right inverses, and so $a$ and $b$ are retracts of each other. But for any object $a$, there are only an essentially small set of objects that are retracts of $a$, and this contradicts the essential largeness of $E$.
Best Answer
Edit : I should clarify that I've interpreted "Etale topos" to mean the petit/small étale topos everywhere. What I've said about Grothendieck-Galois duality only apply to the petit étale topos. If you are talking about the Gros topos, then these part no longer holds. I actually don't know if the Gros étale topos of a fields has a boolean category of coherent object or not.
Yes. I can't give you a proof because as far as I'm concerned this is the definition of the coherent topology. If you see a different definition, maybe edit your question!
Essentially no. Take for example a Boolean algebra $B$. It can be seen as a coherent category (I see $B$ as a poset, and every poset as a category in the usual way). Then the associated topos is the topos of sheaves over the Stone spectrum of $B$, and unless $B$ is finite it has plenty of open that are not also closed (in fact the open that are complemented corresponds exactly to the element of $B$). The general case looks like this though: a Boolean coherent category will gives a topos that "looks like" a Stone spectrum.
The answer is yes for the first half, no for the second, but only because there are very few coherent boolean topos. I would say, a coherent topos is essentially never Boolean (the only exception being the framework of Galois theory): A coherent topos has always enough points, and it can be proved that a boolean topos with enough points is "atomic", that is a disjoint union of topos of the form $BG_i$ where the $G_i$ are localic group. (Here $BG$ is the topos of sets endowed with a continuous action of the localic group $G$.) Adding back the fact that we want this topos to be coherent, we get that the Boolean coherent toposes are exactly the toposes that are finite coproducts of $BG_i$ where the $G_i$ are profinite groups. The second condition you ask for doesn't hold if some of the $G_i$ are non-discrete though. If $G$ is a profinite group (take $G = \mathbb{Z}_p$ for example), then a coherent $G$-set $X$ is a finite $G$-set, so there is going to be an open normal subgroup of $G$ that stabilise all the points of $X$, and the things you get as subobjects of $X^n$ will all be stabilised by the same subgroup.
Yes. The étale topos of any affine scheme is coherent. (For a general scheme it is locally coherent. I'll let an algebraic geometer give you the precise condition under which we get coherence.) In fact, thanks to Grothendieck Galois duality, the étale topos of a field is one of the rare examples of Boolean coherent toposes: it is $BG$ where $G$ is the absolute Galois group of the field, with its profinite topology.
Answering some of the follow up question in the comments.
I would recomand to double checking what I'm going to say here if you plan on using it - I haven't looked at it in enough to details, but I think toposes associated to boolean coherent categories can be characterized as the coherent toposes in which coherent subobject of coherent object have complement. A topos satisfying these condition is clearly the topos of coherent sheaf on a booleancoherent category (by taking all coherent objects) but the converse also seems true. The other condition you have (every object of $C$ is a subobject a power of some fixed object $X$) should corresponds to have a coherent "pre-bound".
The class descriebd you in (1) (including both conditions) are the topos that classyfies single sorted "boolean" first order theory. This additional condition of having a coherent prebound is not automatic at all as the exemple of the topos $BG$ for $G$ a non discrete profinite group shows.
Yes. étale topos of fields are Boolean topos (from their explicit descrition given by Galois duality) so in particular coherent subobject have complements.