We discuss this in more detail in our Analytic Stacks course. The functor $f_!$ exists for all $!$-able maps of analytic rings, which includes $(R,R)_\blacksquare\to (S,S)_\blacksquare$ whenever $S$ is a finitely generated $R$-algebra. It satisfies all the same properties as in the case where both are of finite type over $\mathbb Z$; this can be proved, for example, by "noetherian approximation" (the functor $f_!$ commutes with any base change).

@DavidBen-Zvi Indeed, global Langlands should be about matching objects in (the (restricted) tensor product of) the categories that are identified under local Langlands. There ought to be some global analogue of the Fargues--Fontaine curve, and a moduli space of $G$-bundles on this global object; this should map to the (restricted) product of the moduli spaces of $G$-bundles on the local curves. On the automorphic side, the intended object is the $!$-pushforward of the constant sheaf along this map. On the Galois side, there is a similar picture with spaces of global and local $L$-parameters.

As Dustin quite explicitly says in the first lecture of the course on analytic stacks (and maybe I also say something like this in my first lecture), my primary motivation for developing this theory of analytic stacks is to make something along those lines work. It will, however, look a bit different from what you describe. My ICM report outlines a little bit of this philosophy of "shtukas over $\mathrm{Spec}(\mathbb Z)$". My current work on real local Langlands gives the piece at the real numbers.

The functor $\mathrm{Ch}(\mathbb Z)\to \mathrm{Ch}(\mathbb Z)[w^{-1}]$ is in fact lax symmetric monoidal. This is proved in my paper "On topological cyclic homology" with Thomas Nikolaus, see Theorem A.7.

My guess would be that it is easy to prove that they are not initial, but I didn't try.

Yes, for formal reasons cohomology is dual to homology, and if it is finite-dimensional in each degree, you can reverse this and compute homology as the dual of cohomology. And yes, that $f_!$ is unbounded to the left is a stacky phenomenon. By the way, as the diagonal of $f$ is proper (if $G$ is a compact Lie group), one gets for formal reasons a natural map $f_!\to f_\ast$. This recovers the "norm maps" used in equivariant homotopy theory. The cofiber of the norm map yields "Tate cohomology" which often has some nice periodicity; this can nicely be observed here.

Thanks!! This (e.g. Waelbroeck-Noel's book "Bornological quotients") is precisely the answer I've been looking for. It's interesting that Waelbroeck defined both "compactological spaces" which are the condensed sets that are in some sense "Hausdorff", and also worked on the problem of allowing non-Hausdorff quotients. The difference between condensed vector spaces and his considerations is that in the condensed setup, the passage from Hausdorff to non-Hausdorff objects happens before passing to vector space objects.

This type of question comes up a lot in my work with Clausen; you can ask it over any "analytic ring". Modulo possible technical issues of the specific setup, the answer is "it's true iff Fredholm theory works" in the sense that any trace-class perturbation of an isomorphism has finite-dimensional kernel and cokernel (or, over a general ring, better to say "has perfect cone"). I believe this argument shows that in any of the examples you propose, the answer is Yes. But interestingly, the answer is No for topological $\mathbb C[T]$-modules.