Costello's recent book on renormalization does some of this for QFT, though it's somewhat light on the physical aspects of the subject.

What is a precise formulation of the fully-faithfulness of $\Omega^{\bullet}$? I.e., what are the source and target categories in this assertion?

So you mean imaginary quadratic fields? Then the connection is through the abelianized Galois group, where these things are "easy" (i.e., class field theory). Do you see a way to produce non-abelian extensions with CM elliptic curves? By the way, $\mathbb{Q}$ also has an explicit version of class field theory with $\mathbb{G}_m$ playing the role of the elliptic curve, so your heuristic might indicate that the case of $\mathbb{Q}$ should be even easier than the imaginary quadratic case.

For the Grothendieck spectral sequence, it doesn't matter that your triangulated functors are derived functors of some other functors -- it just gives the spectral sequence relating the cohomologies of the composition to the cohomologies of the second functor applied to cohomologies of the first. This is clear e.g. from the construction is "Theorie de Hodge II." In your setting, a person needs to worry about things like convergence of the spectral sequence, but that's another question!

This is the subject of local cohomology: see e.g. SGA II. The answer (not far from tautologically) is that you need to know (derived) global sections with support in the complement $D$.

I think your last comment isn't quite right: the two things you write give different answers if $\frac{1}{2}\in R$, since then $A/2A=0$.