Let $X$ be a smooth stack and $\mathcal O_X$ the ring of smooth functions on $X$, i.e. for any smooth $M \to X$, $\mathcal O_X(M \to X) = C^\infty(M)$.

In HigherAlgebra, the derived category $\mathcal D(X)$ is defined to be the differential graded nerve of $\mathrm{Mod}(\mathcal O_X)^\circ$, the full subcategory spanned by the fibrant objects (Note that $\mathrm{Mod}(\mathcal O_X)$ is Grothendieck abelian).

Are there good references for the theory of derived functors in this setting?

Is it known when projection formula and (flat) base change work?

Edit: Let me explain why I think that the second question is a sensible one to ask. (I must admit that I found these sources after I posted that question)

There are some rather general results for the unbounded derived category in Spaltenstein. Propositions (6.18) and (6.20) prove base change and projection formula in the case of ringed spaces (given some finiteness-condition on the ring).

Closer and at the same time further from my setting (topological stacks, but only sheaves of abelian groups and the push-forward has to have finite cohomological dimension) is Lemma (6.5.7) in this article.

One of the applications I have in mind is proving that Fourier-Mukai transformations are closed under composition - the only proof I know relies on base change and the projection formula.