Some comments:

Regarding 1): They are quite different. Johnstone actually uses a very general notion of "cover" in his sequential topos -- his site is a full subcategory of metrizable profinite sets (=$\aleph_1$-small profinite sets=countable limits of finite sets=sequential Pro-category of finite sets), but not all of his covers are covers in the condensed/pyknotic sense. I'm not sure the more general covers he allows are of much relevance for his positive results, but they preclude his topos from having enough points. (All the other choices have enough points.) It also means that the generating object $\mathbb N\cup\{\infty\}$ is not actually a quasicompact object in his topos.

So a better comparison would be between the version of Johnstone's topos that restricts to the finitary covers. This admits a geometric morphism from $\aleph_1$-condensed sets. Now $\aleph_1$-condensed sets actually admit a description very similar to this version of Johnstone's topos, but replacing $\mathbb N \cup \{\infty\}$ with the Cantor set, which is the universal metrizable profinite set (i.e. surjects onto any other). But the Cantor set is much bigger than $\mathbb N\cup\{\infty\}$! This has some important consequences, for example $[0,1]$ is quasicompact in $\aleph_1$-condensed sets, but very much fails to be so in Johnstone's topos. Such quasicompactness is used all over the place in our arguments. For example, an extremely important property is the following:

You can consider CW complexes as ($\aleph_1$-)condensed sets. Now any topos has its inherent notion of cohomology, so you can take the resulting cohomology of CW complexes. Then, in the condensed world:

> Applied to CW complexes, the internal notion of cohomology agrees with singular cohomology.

I believe this would fail in Johnstone's topos (correct me if I'm wrong!). And I hope you agree that this is a very desirable property. It's the starting point for seeing that the internal notion of group cohomology of all sorts of topological/condensed groups agrees with the various (ad hoc!) notions of continuous group cohomology you can find in the literature.

On the other hand, almost everything we do in condensed sets could also be done already with $\aleph_1$-condensed sets; in fact, I'm contemplating switching to that setting for some things. One nasty issue is that while condensed abelian groups have enough projectives, there are no nontrivial ones that are *internally* projective. But in $\aleph_1$-condensed abelian groups, $\mathbb Z[\mathbb N\cup\{\infty\}]$ *is* internally projective!

Regarding 2): As you observe, the theory basically works for any $\kappa$. One thing one might like is that as you increase $\kappa$, the pullback functors are fully faithful. While I don't know whether that's always true, it's true at least when the cardinals are either regular or strong limits. And the reason for choosing strong limits is that in that case, one has enough compact projective objects (the extremally disconnected profinite sets), which are very useful (even if not ultimately necessary) for building the theory.

Regarding 3): The main reason is the desire to avoid artificial choices. Let me elaborate by switching to the next question:

Regarding 4): One thing we prove early on is a general Pontrjagin duality on locally compact abelian groups (even on the derived level). But this requires that there are as many discrete abelian groups as there are compact abelian groups. If you work with $\kappa$-condensed abelian groups, the Pontrjagin duality would force one to restrict not only to $\kappa$-small compact abelian groups, but also to $\kappa$-small discrete abelian groups.

Also, if you really want to say that compactly generated topological spaces embed into condensed sets, without implicitly actually talking about $\kappa$-compactly generated ones, again you need to go this colimit over all $\kappa$.

But in practice, mostly everything is $\aleph_1$-compactly generated, and you can just work with $\aleph_1$-condensed sets (or the much larger category of $\beth_\omega$-condensed sets, where you have enough compact projectives).

But as I said above, you absolutely cannot work with Johnstone's topos, the space $\mathbb N\cup\{\infty\}$ is just too small.