My short answer, is that in the great majority of situation where this $\mathcal{Y}/f^*$ plays a role, it is a mistake to see it as a topos. There are exceptions, but most of the time it is not meant to be a topos, but rather a $\mathcal{X}$-indexed topos.
To clarify the discussion, I will follow Joyal's terminology:
I'm calling "Logos" what we usually call a Grothendieck topos. Morphisms of logos are the continuous left exact functor, i.e. the $f^*$.
Toposes are the object of the opposite category of the category of logos. If $\mathcal{X}$ is a topos I denote by $Sh(\mathcal{X})$ the corresponding logos, which I think of as the category of sheaves of sets over $\mathcal{X}$. If I say "$x \in \mathcal{X}$" I mean that $x$ is a point (or maybe a generalized point) of $\mathcal{X}$.
This is mean't to mimic the picture of the conection between frames and locales (where the frame corresponding to a locales is denoted by $\mathcal{O}(\mathcal{X})$.
Now, The topos corresponding to the logos $Sh(\mathcal{Y})/f^*$ is the following object:
$$ \left(\mathbb{S} \times \mathcal{Y} \right) \coprod_{\mathcal{Y}} \mathcal{X}$$
where $\mathbb{S}$ is the Sierpinski topos, i.e. corresponding to the topological space with one closed point and one open point. (this is easily seen using that colimits of toposes corresponds to limits of logos, which are justs limits of categories)
So you can think of it as a kind of "cone construction" where the Sierpinski space is used as an interval. This object can indeed be interesting, and you can somehow guess that it will be especially interesting for the theory of local morphisms as you mentioned in your post.
But, in my opinion, this topos as simply nothing to do with the idea of working with $\mathcal{Y}$ as "an object over $\mathcal{X}$", i.e. to try to think of the map $\mathcal{Y} \rightarrow \mathcal{X}$ by somehow looking the fiber $\mathcal{Y}_x$ for $x \in \mathcal{X}$ and how it varies when $x$ varies in $\mathcal{X}$. Which is what we want to do in all the case you mentioned:
A proper map is a map whose fiber are compacts, in a "nicely locally uniforme way""
A locally connected map is map whose fiber are locally connected in a nicely locally uniform way.
A separated map... well you can actually also see it as a map whose fiber are separated in a locally uniforme way, but that is not quite what the definition you gave really says. The way I think about this definition is that when you have a class of map stable under composition and pullback then it is nice to consider the class of maps whose diagonals have this property, because you then get for free the lemma that "if $f \circ g$ is proper and $f$ is separated then $g$ is proper".
Let's now look at what happen when you want to work with $\mathcal{Y}$ as an object over $\mathcal{X}$ :
Essentially, instead of looking at $Sh(\mathcal{Y})$ you want to look at something like $Sh(\mathcal{Y}_x)$ for all $x \in \mathcal{X}$, which should give you some kind of family of logoses parametrized by $x \in \mathcal{X}$.
In the same way that the correct notion of "familly of set parametrized by $x \in \mathcal{X}$ is a sheaves over $\mathcal{X}$ the correct notion of such familly of logos is something like "a sheaves of logos". It is not quite just a "sheaves in the category of logos" because the definition of logos involved some infinitary operations (the infinite coproduct/colimits) which you want to replace by $Sh(\mathcal{X})$-indexed colimits/coproducts. So the correct notion is what I would call an "internal logos" and is a special kind of sheaf of logos (It is exactly a sheaf of logos which admits $Sh(\mathcal{X})$-indexed disjoint and universal coproducts).
Also note that these sheaf of logos in particular gives you the type of structure that you asked about in your last question. (they satisfies stronger property though, like Beck Cheavely conditions related to the fact that they have indexed colimits)
Then for technical reason, we tend to look at sheaves of categories rather as indexed categories or fibered categories, that is why you end up with a fibration over $Sh(\mathcal{X})$.
But if you somehow forget that it was mean't to be a "sheaf of logos" over $Sh(\mathcal{X})$ and see the total category of the fibration as a new logos, then you just get a completely different and new object that have very little to do with what I was describing.
I'm finishing with an informal discussion of why the $\mathcal{X}$-indexed logos corresponding to $\mathcal{Y} \rightarrow \mathcal{X}$ should be $Sh(\mathcal{Y})/f^*$. This is not a proof, just some kind of heuristic arguement. The proof is justs that there is a relatively deep theorem saying that the category of $Sh(\mathcal{X})$-indexed logoses is equivalent to the category of toposes over $\mathcal{X}$ and that the equivalence is given by this construction, and is compatible with pullbacks along maps $\mathcal{X'} \rightarrow \mathcal{X}$.
So What should be the "sheaf of logos" (or internal logos) corresponding to $\mathcal{Y} \rightarrow \mathcal{X}$. I need to think about "what should be its section over some étale cover $p: \mathcal{E} \rightarrow \mathcal{X}$".
I want something that, in a continuous way associate to each $e \in \mathcal{E}$ a sheaves over $\mathcal{Y}_p(e)$, i.e. something that continuously in $e \in \mathcal{E}$ and $y \in \mathcal{Y}_{p(e)}$ associate a set. So basically it is a sheaf of set over $\mathcal{Y} \times_{\mathcal{X}} \mathcal{E}$.
Now if $\mathcal{E}$ is the etale space of a sheaves $ E \in Sh(\mathcal{X})$, one has that $Sh(\mathcal{E}) = Sh(\mathcal{X})/E$, and $Sh(\mathcal{Y} \times_{\mathcal{X}} \mathcal{E}) = Sh(\mathcal{Y})/f^* E$
So in the end you do get the fibered category we are talking about.
In my view, the correct notion of "sheaf of Xs" is "internal X in the topos (or $\infty$-topos) of sheaves of sets (or spaces)". (I mentioned this previously on MO here.) Since sheaves of sets are a limit theory, if X is also defined by a limit theory (i.e. the category of Xs is locally presentable), then by commutation of limits this is the same as a sheaf of Xs in the naive sense. But for other values of X it gives different answers. In fact, the answer it gives may depend on exactly how the theory of X is presented; but that's reasonable becaues sometimes there is more than one correct notion of "sheaf of Xs" (equivalently, there is more than one version of X in the internal constructive logic of a topos). For instance:
- If X = fields, there are discrete fields, Heyting fields, and residue fields. I think discrete fields are the one that corresponds to viewing fields as models of a limit-colimit sketch (i.e. as an accessible category), but the others are often more useful (e.g. Heyting fields include the sheaf of continuous real-valued functions on a topological space).
- The case of X = Kan complexes has already been mentioned in other answers. Although in general once you're talking about homotopy theory, it's better to incorporate the homotopy theory into the ambient $\infty$-topos and work with stacks.
- If X = finite abelian groups, there are different notions of finite object in a topos.
- If X = topological spaces, you can internalize that directly, but often more useful is to internalize the notion of locale -- for instance, a "sheaf of locales" on a sufficiently nice topological space $Y$ is equivalent to a space over $Y$.
- If X = local rings, written as a geometric theory, this definition gives you the generally accepted definition of "sheaf of local rings", i.e. a sheaf of rings whose stalks are local.
This definition of "sheaf of Xs" satisfies your criteria (3) and (5). It also satisfies your criterion (1) in as strong a way as I think could be expected: the category of internal Xs in a topos behaves exactly like the ordinary category of Xs, as long as the latter is interpreted using constructive logic. And it satisfies your criteria (2), (4), and (6) if the theory of Xs is geometric, hence has a classifying topos -- which I think is the most general situation in which one can expect these properties to hold.
(Note, by the way, that your criterion (6), as well as the stronger version referring to all presheaf toposes, is a special case of your (4), since presheaves on $C$ are the Cat-enriched copower of the terminal topos by $C$ in the bicategory of toposes.)
Best Answer
There is indeed a strong analogy between this situation and adjoining a polynomial variable to a ring, except that the direction of the arrows is somewhat messed up:
A ring homomorphism $\mathbb{Z}[X] \to R$ is the same thing as a ring homomorphism $\mathbb{Z} \to R$ (of which there is exactly one) together with one arbitrarily chosen element of $R$. (More precisely, we have a bijection, natural in the ring $R$.)
A geometric morphism $\mathcal{E} \to \mathrm{Set}[\mathbb{T}]$ is the same thing as a geometric morphism $\mathcal{E} \to \mathrm{Set}$ (of which there is exactly one, up to unique isomorphism) together with one arbitrarily chosen $\mathbb{T}$-model in $\mathcal{E}$. (More precisely, we have an equivalence of categories, natural in the topos $\mathcal{E}$.)
In case you wonder how to adjoin a $\mathbb{T}$-model to another topos than $\mathrm{Set}$, you can simply take the product $\mathcal{E}[\mathbb{T}] = \mathcal{E} \times \mathrm{Set}[\mathbb{T}]$, just like we have $R[X] = R \otimes \mathbb{Z}[X]$. (But be aware that the product of toposes is not given by the product of the underlying categories.) (And one could generalize to the case where $\mathbb{T}$ is not an ordinary geometric theory but instead a geometric theory internal to $\mathcal{E}$.)
In case you wonder, if you can adjoin a model of an arbitrary geometric theory to a topos, then what else can you adjoin to a ring apart from just a new ring element: you could for example "adjoin" two elements $X$ and $Y$ with the property that $X^2 = 5Y$; this would yield $R[X, Y]/(X^2 - 5Y)$. But the analogy arguably starts breaking down here.
The fact that the geometric morphisms go in the opposite direction from the ring homomorphisms (and that we use the product of toposes where we used the tensor product (coproduct) of rings) can be explained by saying that toposes are "geometric" objects and rings are "algebraic" objects. Or alternatively by the observation that while an element of a ring can be "pushed forward" along a ring homomorphism, a $\mathbb{T}$-model in a topos can be pulled back along a geometric morphism.