Perhaps the Duflo isomorphism is what you're looking for.

If you really mean the word "coherent" in the title, then the answer is no. E.g., you would want a $j_!$ morphism for $j$ an open embedding, but one can see that there's no left adjoint to $j^*$ in this setting (since it would imply tensor products commute with infinite products).

Not quite: you really need to ask that the underlying $\mathcal{O}_{X\times Y}$-module be a vector bundle. If $Y$ is a point, then a $p^*D_Y$-module is just the same thing as a quasi-coherent sheaf on $X$.

I think at least the Hodge filtration should be. The weight filtration is of a somewhat more Betti-origin, so I wouldn't be so surprised to find an example where it's not geometric in this way. But it's quite non-trivial to extract anything like this from Saito -- the set-up uses polarizations (and therefore the real numbers) in a quite non-trivial way.

@JSE: Could you make that analogy more precise?

Is your question: what improvements does it give to Deligne's contruction? Otherwise, it's clear: motives are very constrained, e.g., their Frobenius eigenvalues when restricted to primes of good reduction. This is how Deligne proved Ramanujan's conjecture, though he didn't quite produce the motive. But after that, I think the question of being motivic is very natural.

In fact, there's a whole book! amazon.com/Symmetry-Monster-Greatest-Quests-Mathematics/dp/…

Ah, I'm sorry. I misunderstood what "any $V$" meant. Well, the following general theorem answers your first question (positively): an indecomposable vector bundle on a smooth projective curve over $\mathbb{C}$ admits a flat connection if and only if the vector bundle is semi-stable. A partial answer to the second: adding a non-zero 1-form to a connection on $E$ gives different holonomy since global functions on $E$ are constants. I don't see what operators can arise from these particular bundles.

As for any complex manifold, gauge equivalence of holomorphic flat connections is determined by the holonomy. In this case, this means gauge classes are the same as isomorphism classes of finite dimensional vector spaces equipped with two commuting operators (and these isomorphism classes are easy to compute via Jordan form).

Supposedly it takes some constrained values at $0$.

If $X$ and $S$ are finite type over $\operatorname{Spec}(\mathbb{Z})$, then what you've said is indeed well-defined and is just the zeta function of $X$.

What is the shape of an answer you are hoping for? E.g., should it be a holomorphic function? For "relative" algebraic geometry, you typically want something of local nature on the base, and given the form of usual zeta functions (which you seem to want as the result when $S=\operatorname{Spec}(\mathbb{Z})$), it's hard to see what form it would take.

Ah. Perhaps a less hands-on proof is to show that the datum of equivariance is equivalent to the datum of descent datum (hint: $X\underset{X/G}{\times}X\simeq G\times X$). Then just apply faithfully flat descent. There's a generalization of this claim for "bad" actions of $G$ on $X$: it says that $G$-equivariant quasi-coherent sheaves on $X$ are the same as quasi-coherent sheaves on the stack $X/G$. This is basically formal once you have the definitions and the case of principal bundles. It also immediately implies the statement you specified about $G$ and $H$.

Half of your instincts are correct: you can't recover the zeta function from what happens at $\mathbb{C}$. You can see this already by just considering Dedekind zeta functions: certainly they carry more information than the degree of the extension. The zeta function is rather something of an amalgamation of what happens at every prime, or a kind of generating function for the number of solutions to some equation or what have you.