Existence of a nice-ish topology on the powerset of a topological space

Question

This is a follow-up question to my previous question, Existence of a *really* nice topology on the powerset of a topological space, which, in a few words, asked about whether given a topological space $X$ we could induce a compatible topology on its powerset $\mathcal{P}(X)$ in such a way that all the usual operations from set theory (unions, intersections, differences, etc.) become continuous with respect to this topology.

In the hope to make this question more self-contained, I'll repeat the relevant conditions stated there:

1. The map $\iota\colon X\to\mathcal{P}(X)$ given by $x\mapsto\{x\}$ is continuous.
2. Binary union ${\cup}\colon\mathcal{P}(X)\times\mathcal{P}(X)\to\mathcal{P}(X)$ is continuous.
3. Binary intersection ${\cap}\colon\mathcal{P}(X)\times\mathcal{P}(X)\to\mathcal{P}(X)$ is continuous.
4. Difference ${\setminus}\colon\mathcal{P}(X)\times\mathcal{P}(X)\to\mathcal{P}(X)$ is continuous.
5. Arbitrary union ${\bigcup}\colon\mathcal{P}(\mathcal{P}(X))\to\mathcal{P}(X)$ is continuous.
6. If $f\colon X\to Y$ is a continuous map of topological spaces, then so are its direct and inverse images \begin{align*} f_{*} &{}\colon\mathcal{P}(X)\to\mathcal{P}(Y),\\ f^{-1} &{}\colon\mathcal{P}(Y)\to\mathcal{P}(X). \end{align*}

There, James Hanson proved the following inexistence result, which I paraphrase:

Theorem. Let $X$ be a topological space. If $(\mathcal{P}(X),\tau_1)$ and $(\mathcal{P}(\mathcal{P}(X)),\tau_2)$ satisfy conditions 1–6 and $\tau_1$ is not indiscrete, then $X$ is discrete.

Now, looking at the case of the Vietoris topology (arguably the most commonly used powerset topology), I noticed that condition 7 is misguided: when $\mathcal{P}(X)$ is equipped with the Vietoris topology, the inverse image function $$f^{-1}\colon\mathcal{P}(Y)\to\mathcal{P}(X)$$ is continuous as a relation iff $f$ is closed and open, and the continuity of $f$ is in fact irrelevant. Meanwhile, the continuity of $f$ still implies that the direct image $f_*$ of $f$ is continuous, but again only as a relation. (See the references here for an explanation of the not-so-well-known notion of continuity of relations).

Thus, instead of condition 6 above, a more reasonable set of conditions would be to ask that just the direct image $f_*$ is continuous (either as a relation or ideally as a function).

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Question I. Is there an interesting topology on $\mathcal{P}(X)$ satisfying conditions 2–4? What about 1–4?

Question II. Moreover, is there a finest such topology in the sense of being a maximal element in the sublattice of all topologies on $\mathcal{P}(X)$, ordered by inclusion, spanned by those satisfying 2–4 (resp. 1–4)? Must such a topology be in fact a maximum element of this sublattice?

P.S. The minimal such element is the indiscrete topology.

Question III. Is there a general procedure of inducing interesting topologies on powersets (like the definition of the Vietoris topology) in such a way that 1–5 hold, and also $f_*$ is continuous¹ whenever $f$ is so?

(¹Either as a relation or as a function, I'm interested in both questions)

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Lastly, we recall the following two constraints on any topology satisfying 2–4, proved by James Hanson.

Proposition. If $\mathcal{P}(X)$ satisfies 2–4, then it is a homogeneous topological space.

Proposition. If $\mathcal{P}(X)$ satisfies 2–4, then $\overline{\{\emptyset\}}$ is a filter of $(\mathcal{P}(X),\supset)$, so that:

1. If $B\in\overline{\{\emptyset\}}$ and $A\subset B$, then $A\in\overline{\{\emptyset\}}$.
2. If $A,B\in\overline{\{\emptyset\}}$, then $A\cup B\in\overline{\{\emptyset\}}$.

Moreover, $X$ is indiscrete iff $X\in\overline{\{\emptyset\}}$.