The value of that thing at $X$ wouldn't be the spectrum of the ring at $X$; rather, the spectrum is some kind of total space over $\mathcal C$ and the value at $X$ are sections of it. These have the correct variance.

Both notions of cohomology take filtered colimits to cofiltered limits, so one can reduce to the case of finite CW complexes.

Topology with closed covers: You have to allow in addition any surjective selfmap of the Cantor set. Cohomology of CW complexes: Lecture 3 of Condensed Mathematics

It's similar, but If I see it right, then they allow too few covers (only the open covers). In the context of your other question, this would mean that gluing along closed covers doesn't work as expected, but gluing along closed covers is required for geometric realization to work as expected.

Yes. The condensed structure it acquires gives the constructible topology of $\mathrm{Spec}(A)$. To see the actual Zariski topology, it is then sufficient to remember the specializations, recorded in the poset structure.

Yes, that sounds right.