This seems only to work in the case when $M$ is generated by global sections (e.g., the D-affine case).

For your last two questions: 1) a D-module is a vector bundle with connection if and only if the trivial filtration (with a jump in only one degree) is a good filtration, so yes, and 2) $\mathbb{A}^2\setminus\{(0,0)\}$ is not D-affine.

I should add: regular functions on a loop group naturally form a Tate space -- this is so for sufficiently nice ind-schemes (though ind-infinite type is allowed, and accounts for half of the semi-infinity of the Tate space). And the completed symmetric algebra of $\mathfrak{g}((t))$ forms a topological coisson algebra in Beilinson's sense.

Perhaps the definition of "topological coisson algebra" from Beilinson's "Remarks on topological algebras" is relevant for you. It's the definition of Poisson algebra in the setting of Tate vector spaces.

Yes: it is due to Artin and Mazur.

Yes, it carries the discrete topology.

My answer was meant as a bit of a joke: certainly one requires hypotheses for such representability theorems. The one I know is called "presentable." But anyway, it's a precise statement for vector spaces for any field $K$. You topologize the dual as a subset of $\prod_{v\in{V}}{K}$, where the latter has the usual ("Tychonoff") topology (and the map $V^*\to\prod_{v\in{V}}{K}$ is $\lambda\mapsto (\lambda(v))_{v\in{V}}$).

Continuity is equivalent to representability (in both cases).

I think your formula isn't quite right: consider the case $X=S^0$. The problem is that $\Omega\Sigma X$ is group-like while $\overline{X}$ is not. But it's direct to show that the group completion of $\overline{X}$ is indeed computed by $\Omega\Sigma X$.

It also seems relevant to ask when phrases like "variety over $K$" (which naturally leads to "scheme $X$ over $S$" and the accompanying picture) started to be used.

Illusie says he thinks Grothendieck introduced the vertical arrow: math.uchicago.edu/~mitya/langlands/reminiscences1.pdf.

is equal to the limit of $K\to\mathcal{C}$.

I only know the quasi-categories model, but as long as you have a notion of limit between functors of $\infty$-categories you're okay. If you have $K\to \mathcal{C}$ a map from a simplicial set $K$ to a quasi-category, take a categorical equivalence $K\simeq J$ where $J$ is a quasi-category with a functor $J\to\mathcal{C}$ such that the diagram commutes. Then, using whatever equivalence between quasi-categories and you're other model, transfer $J\to\mathcal{C}$ to some $J'\to\mathcal{C}'$ in your other world and take the limit. The point is that the limit of $J\to\mathcal{C}$...

One general form is that $GL_n$-torsors in the fppf (or a fortiori etale) topology coincide with Zariski $GL_n$-torsors (alias: rank $n$ vector bundles). Your first statement follows, since the left hand side computes isomorphism classes of etale $GL_n$-torsors on $\Spec(K)$, which then coincides with isomorphism classes of rank $n$ vector bundles on $\Spec(K)$, which is obviously just the one. Your second statement (which is a little imprecise, but presumably your $H^1$ is etale) is this for $n=1$.

I'm not quite sure what it would mean for a definition to be independent of the model. But Lurie's book proves that limits and colimits are invariant under categorical equivalences in the "$K$-variable," which I think answers your question. However, sometimes in the context of quasi-categories, it's more convenient to allow an arbitrary simplicial set, just like sometimes when doing homotopy theory with simplicial sets it's convenient to allow non-Kan sets. E.g., the quantity of generators and relations is much smaller.

What are the precise bounds of your question? Are you asking what invariants can be read off of the triangulated category alone? Or are there certain distinguished objects, like the constant sheaf?