But at the end of the day, one should be able to write down for every $\mathcal{L}$ an "explicit" category $C$ (described e.g. as a certain category of Whittaker sheaves) with a map from $C$ to the category of $\mathcal{L}$-eigensheaves. (And maybe one can do even more... but at this point already I would need to assume multiple wide open conjectures to even formulate my statements. And I am having difficulty expressing what I want to say in the space of the comment box.)

that $\text{D-mod(Affine flag variety)}$ corresponds to $\operatorname{QCoh}(\hat{b}/\hat{B})$ recovers the tame (w unipotent monodromy) statements. There are even some correspondences which get into arbitrary ramification, e.g. that $g((t))-\operatorname{mod}_{crit}$ should correspond to $\text{QCoh(Opers)}$, and from that you can extract (rather painful) arbitrary ramification statements. Admittedly the original question was in the $l$-adic sheaf setting, where I believe that roughly the same story should hold but I am scared to write down any concrete statements...

@WillSawin In the categorical geometric setting (so char 0 & with D-modules) the strongest statement should be (modulo issues of non-temperedness) that $\operatorname{D-mod}(\operatorname{Bun}(X-S))$ and $\operatorname{QCoh}(\operatorname{LocSys}(X-S))$ correspond to each other under local categorical geometric Langlands. Of course, to extract concrete statements from this you need to pin down exactly what local geometric Langlands is. E.g., the knowledge that $\text{D-mod(Affine grassmannian)}$ corresponds under LGL to $\text{QCoh(pt/}\hat{G})$ recovers the unramified statement, the knowledge

This statement should be true but should also be far from the strongest possible conjecture.

The critical shift can be eliminated by twisting w.r.t. a cetain line bundle. Then the only fact to note is that the D-modules on the LHS are automatically regular holonomic - see e.g. 11.6 of Hotta-Takeuchi-Tanisaki for this (technically they do it only in the finite-dimensional case, but it is easy to deduce what you want from that). In general I think the resolution to these issues is more transparent to see in the Beilinson-Bernstein context, so maybe thinking about that first may be helpful.

There are many examples involving cubic fourfolds - see arxiv.org/abs/1601.05501.

A comment: I believe that the Lovasz local lemma gives a positive answer in the case where $m\geq 7$; this is weaker than Seva's bound but seemed worth mentioning in case it may be useful in a different scenario.

@Cutthewood Seeing the topos as the fundamental object rather than the site is certainly a conceptual advance, and I can't say whether or not Grothendieck had this idea in mind from the very start (though if he didn't, he developed it soon after.) At least in SGA4, the emphasis is already on topoi and not just sites. I suppose if you interpret the question as "What led Grothendieck to consider the notion of topos as more fundamental than the notion of site?" this is an interesting question...

@Cutthewood But the question is "how were they invented" - and the answer is "they were invented at the same time as sites, because it is hard to talk about sites without also inventing topoi."

I'm not sure I understand the question, in particular the "(as opposed to site)" part. My impression is as follows: The reason sites were invented was to be able to define etale cohomology as the derived functor of global sections on the category of etale sheaves. Therefore the category of etale sheaves (the topos) was in play from the very start. I assume your comment about functoriality for crystalline cohomology refers to the existence of morphisms of topoi which don't come from morphisms of sites, which I see as a separate phenomenon...

The proof you linked to works for all uncountable algebraically closed fields of characteristic $0$. I think you need a new idea to extend it to say, $\overline{\mathbb{Q}}.$ For (uncountable algebraically closed) fields of characteristic $p$, this seems closely related to asking if $M_g$ is unirational for large $g$, and to my knowledge this is still open (see the final remarks in arxiv.org/abs/1702.04404).

See mathoverflow.net/questions/296682/… for your first question.

The arithmetic analogue would be a simple construction that takes in some kind of object (a galois rep/motive with extra structure) and spits out a representation of a p-adic group. Looking at this description, it is evident what is missing in the arithmetic setting: an analogue of the category of modules over the affine Lie algebra. This is probably the most important tool available in the geometric setting that so far does not seem to have been transferred to arithmetic.

I find the local version of this story a little more transparent: we have a (categorical) representation of the loop group for every oper on the punctured disk. This is because the universal enveloping algebra of the affine Lie algebra has a large center equivalent to the algebra of functions on the space of opers, and so for every oper we get a character of this center. Then our categorical representation is just the category of affine Lie algebra modules where the center acts by this character.

@Lycurguscup This difference is very much present in geometric Langlands. In the geometric setting, unipotent representations are more or less completely understood, while general tamely ramified (w/ the definition from number theory) representations haven't received that much attention at present. Furthermore, the unipotent representations seem to be more directly relevant to understanding "classical" geometric representation theory. My guess is that because there hasn't been much GRT work on general tamely ramified reps, "tamely ramified w/ unipotent monodromy" started getting abbreviated.

Just to make it explicit that this answers Q1 as well: If none of the $r$ are elements of $\mathbb{F}_p$, then the only way $x^3-ry^3-z^3$ can be zero mod $p$ is if $y=0$, i.e., if you are at the singular point mentioned in the question. The roots of $t^4-t^3+t^2-t+1$ are the primitive $10$th roots of unity, so it will have no roots mod $p$ exactly when $p$ is not congruent to $1$ mod $10$.