Comment about 3: From the analytic point of view the fact that top forms carry a \emph{right} action of $D$ is very natural. Namely, there is a pairing between top forms and functions given by $(\omega,f)\mapsto \int f\omega$ (if the space is not compact, we'd need to say something about behavior at infinity to ensure convergence). If you think of top forms as functionals on the space of functions (i.e., distributions), it is clear where the right action of $D$ comes from.

In case you are interested, there is a conceptual explanation that was not mentioned here so far. $D$-modules are the same as crystals of $O$-modules (or, if you prefer, quasi-coherent sheaves on the de Rham stack), which makes the tensor structure obvious. This is similar to the differential-geometric question/answer: Why does a tensor product of two bundles with a flat connection carry a flat connection? Answer: Because flat connection can be viewed as the data of parallel transport in fibers of a vector bundle (making the bundle into a local system), which makes tensor product clear.

I don't see why you are skeptical about the closed monoidal structure. Let $M,N,P$ be three $D$-modules. The $D$-module morphisms $M\otimes N\to P$ are simply $O$-bilinear maps $\phi:M\times N\to P$ that satisfy the Leibniz rule $\tau\phi(m,n)=\phi(\tau m,n)+\phi(m,\tau n)$ for any vector field $\tau$. If we fix $m\in M$ and consider $\phi(m):N\to P:n\mapsto\phi(m,n)$, the condition becomes $\phi(\tau(m))=[\tau,\phi(m)]$. The right-hand side is the action of $\tau$ on $\phi(m)\in Hom(N,P)$, so this is exactly the condition that $\phi$ is a $D$-module morphism $M\to Hom(N,P)$.

Are you talking about rigid monoidal category or about closed monoidal category here?

I don't have a reference, but it is pretty straightforward: enough to consider the case $S=Spec(R)$, and then a coherent sheaf $F$ is given by a finitely generated graded $R[x_0,\dots,x_n]$-module $M=\bigoplus M_d$; for large enough $D$, $M_D$ generates $M_{\ge D}:=\bigoplus_{d\ge D} M_d$, so the annihilator ideal of $M_D$ in $R$ is contained in the annihilator ideal of $M_{\ge D}$.

The claim is that for any (quasi-compact noetherian) scheme $S$ and any coherent sheaf $F$ on $S\times{\mathbb P}^n$, we have $p(supp(F))=supp(p_*(F(n)))$ for $n\gg 0$. Here $p$ is the projection onto $S$.

@R.vanDobbendeBruyn: to me, it seems that adding the trivial representation is simply about restating general position in V considered as an affine space as a general position in the linear space $(V\oplus C)$. (Or if you like, $V$ as an affine space is considered as a subset in the projective space of lines in $V\oplus C$.) Since the question already asks about projective space corresponding to V, this does not seem to be related...

Doesn't this basically show that $F$ is not the direct limit of $F_n$ in the category of presheaves (which was required in the question)?

As მამუკა ჯიბლაძე says, the properness is not really an issue: You don't need the properness of $p_1$, but only of the restriction of $p_1$ onto $\Sigma$. As for the missing twist by $\omega$, I think you are right, $p_1^*$ should be $p_1^!$. This can be seen already when $G=\{e\}$: then $\Phi=Id$ (the missing $G$-invariants won't matter), but the given formula for $\Psi$ has a non-trivial twist in it.

If the action of $G$ on $X$ is non-trivial, then the kernel that you describe is not $G\times G$-equivariant. The kernel should be $\bigoplus_g O_{\Gamma_g}$, where $\Gamma_g\subset X\times X$ is the graph of action of $g$ (or, more canonically, it is the direct image of $O$ under the action map $G\times X\to X\times X$).

Note that if $F$ has this property, then so does $F\otimes p_Y^* E$ for any sheaf $E$ on $Y$. In particular, you know that $dim(supp (p_*(F\otimes p_Y^*E)))=0$ for any $E$. Taking $E$ to be a high power of an ample line bundle gives you what you want.

@ BenWebster --- at first, I thought the problem is indeed as trivial as you suggest, but this is not what the question is asking. The goal is to show that we can find N+2 points in general position among those given in the N-dimensional projective space. Here being in general position means that removing any one of them still leaves a spanning set. For example, if N=2, we need four points on a plane such that no three of them are collinear.

I think it is false. For instance, let us say $X$ is a curve of genus at least two, and $E=F$ is a line bundle. Then the only globally defined differential operators $E\to E$ are constants (this can be easily seen because the leading term of a differential operator is a section of $T_X^{\otimes d}$, which automatically vanishes if $d>0$ because of the genus condition). However, if $E$ is very positive, there are many non-constant maps $\Gamma(X,E)\to\Gamma(X,E)$.

@Jason: it seems to me that if the action is linearized, normality is not required. If a torus acts linearly on the projective space, then the closure of any non-zero-dimensional orbit contains at least two fixed points (e.g., if T=Gm, they are lim(tx) when t->0 and when t->infty). This implies the statement.

I'd say it is like the definition of $P^n$, but it is not purely a `matter of style'. This essentially rephrases Mohan's comment, but here it is: On the level of points, there is of course no difference between subs and quotients: we can just think of short exact sequences $0\to A\to E\to B\to 0$. Now as you consider families (say, over $S$), you really want sequences $$0\to A\to E\boxtimes O_S\to B\to 0$$ with $A$ and $B$ flat over $S$ (so you have a family of short exact sequences). But flatness of $B$ implies flatness of $A$, so we can forget $A$ and think of $B$ only.

@YosemiteSam By the way, even when X is an elliptic curve, P does not need to be a line bundle --- any indecomposable vector bundle would do.