Is it not the case that then both sides are zero? LHS is trivial by definition, while for the RHS this follows from BB localization (but probably there is a more direct way to see it).

What part of Beilinson's article are you looking at? As far as I can tell most of that article relies on the stronger hypothesis of existence of a motivic t-structure, which to my knowledge does not easily follow from conservativity.

I'm sure that you can find more canonical-looking presentations in specific cases, but in general I don't know (other than some tautological presentations.) And yes, pushforward along a quasi-compact quasi-separated morphism commutes with filtered colimits (Proof: Reduce to case Y affine. In this case, pushforward is a finite limit, which commutes with filtered colimits.)

Here is how to go from quasi-coherent $F$ to coherent $F$: On any quasi-compact, quasi-separated scheme, quasi-coherent sheaves are filtered colimits of coherent sheaves. Since $\mathcal{O}_Y$ is compact for $Y$ such a scheme, the map $\mathcal{O}_Y^n\rightarrow f_*F$ factors through one of these coherent sheaves, which gives you what you want. The key facts here are the same as those appearing in Stacks Project 27.22 or EGA I.9.4.

Even if you start with a projective variety, I don't know if you can get all that much information about cohomology of arbitrary smooth closed subvarieties of codimension $1$. For instance, let $X$ be the blow-up of any smooth projective variety at a point, and let $E$ be the exceptional divisor. The cohomology of $X$ may be very complicated, but $E$ sees almost none of that complexity.

Well, there are counterexamples such as e.g. $Y=\mathbb{A}^1_{\mathbb{C}}$, $X$ the disjoint union of $\operatorname{Spec}\mathbb{C}$ over all closed points of $Y$. Maybe you want to add some extra conditions on $X$ to avoid this type of scenario.

This is one of the most studied varieties in all of algebraic geometry; a phrase to to google is "nilpotent orbit." See e.g. Ch.3 of Chriss-Ginzburg for an introduction.

See mathoverflow.net/questions/121678/…

You have similar uniformization statements for Shimura varieties and shtukas (as double coset spaces such as $G(K)\backslash G(\mathbb{A})/G(\mathbb{O})$) which gives you a direct link to automorphic forms and tells you that in either case, the cohomology comes with a bunch of Hecke actions.

As a general rule, there are much fewer maps between algebraic varieties than there are between the underlying topological spaces. For instance, the pullback map on cohomology induced by your $f'$ has to be compatible with the mixed Hodge structures on both sides.

The standard proof of this exercise very much relies on smoothness + properness, in that the key point is Serre duality.

If by $D_X$ you mean the ring of Grothendieck differential operators, then I believe the answer is yes (and that this uses no specific properties of $D_X$, only that it is an associative algebra - this is credited to Bernstein in at least one reference). On the other hand, both categories are very different from the "true" category of $\mathcal{D}_X$-modules for $X$ singular, so I'm not sure if this is really what you are after.

See mathoverflow.net/questions/289941/…

Unfortunately I'm not sure if I can phrase the relation without introducing a lot of notation. This is the subject of section 12.8 of Arinkin-Gaitsgory. An important subtlety here to be aware of is that the true category of Hecke operators is larger than the naive one described by Mirkovic-Vilonen and is only visible at the derived level.

then Ramanujan-Petersson should be the statement that the image of D-mod_cusp(Bun_G) under this equivalence should be contained in QCoh(LocSys). 3. However, you would want a purely automorphic (without invoking Langlands) statement, in terms of the Hecke action. For a purely automorphic formulation of temperedness, see section 12.8 of the Arinkin-Gaitsgory singular support paper. So now the question is if D-mod_cusp(Bun_G) is contained in D-mod_temp^x(Bun_G) for all points $x$ of your curve. Anyways, this is getting pretty technical... I can try to clarify individual points if asked.

This is a very interesting question, and there is a lot to be said about it. Here are just a few remarks: 1. The role of eigenvalues is played by local systems, not opers. 2. According to wikipedia, the general statement of Ramanujan-Petersson is that for a globally generic cuspidal automorphic representation, each local component is tempered. Globally generic is irrelevant in the geometric setting, and temperedness corresponds (on the LocSys) side to lying in QCoh(LocSys) rather than IndCoh. So if you assume the statement of geometric Langlands (D-mod(Bun_G) is equivalent to IndCoh_N(LocSys))