My favorite is $Vect_{n,d}(C)$ of vector bundles on a curve of genus $g\geq 2$ of fixed rank and degree up to isomorphism. The automorphism group has positive dimension, hence not a Deligne-Mumford stack. Maybe what you should expect is to replace Deligne-Mumford by Artin.

If I assume the question is asking something along the lines of what information on the cotangent bundle recovers the D-module then this is going to be very hard. For example, to make a statement precise one will need to dive into the depths of the proof of the codimension three conjecture of Kashiwara and Vilonen: front.math.ucdavis.edu/1209.5124. To make things even harder this conjecture, well theorem, only applies to regular holonomic D-modules. I have not heard of any proposed conjecture for non-holonomic D-modules...

I think about the same argument will work for cohomology but instead you should get something like $HH^*(D_M)\simeq H_{DR}^*(M)$ but this time with no shift in the grading. This is like a Poincare duality between Hochschild cohomology and Hochschild homology.

I once had a discussion with an expert in exotic structures on 4-manifolds who said that a group of physicists claimed that the reason for dark matter is exotic structures on $\mathbb{R}^4$. I don't know if anything ever came out of it, but it is an interesting idea.

higher brackets, an explicit $L_{\infty}$-quasiisomorphism, and an $L_{\infty}$-homotopy. But,'wise' might be too difficult to determine.

Hi Damien, Thank you for your reply. The definition of homotopy equivalence that I had in mind was from Fukaya: \emph{Deformation theory, homological algebra, and mirror symmetry} page 44. Then when everything is in the $L_{\infty}$-category quasiisomorphism implies homotopy equivalence (But maybe the converse is not true). My other hope, maybe somewhat naive, was to choose a 'wise' enough homotopy at the chain level and then use an $L_{\infty}$ version of Markl:\emph{Transferring $A_{\infty}$ structures}. If the choice is 'wise' enough then this will imply that $T_{\poly}$ can't have any

Ben, What do you mean by "any such representation much have coisotropic limit." Does this mean that a module over deformation quantization becomes a module supported on a coisotropic subvariety when $\epsilon \to 0$?