I would appreciate either an explanation or a reference for what is going on here.

Motivation:

Let $f : X \rightarrow Y$ be a morphism of algebraic varieties. The derived projection formula implies that for a sheaf $\mathcal{F}$ on $Y$, we have

$$Rf_\ast Lf^\ast \mathcal{F} \cong Rf_\ast \mathcal{O}_X \otimes^L \mathcal{F}.$$

Suppose for example that $R^i f_\ast \mathcal{O}_X = 0$ when $i>0$, and is $\mathcal{O}_Y$ when $i=0$. It's natural to guess that the cohomology of the right hand side is just $\mathcal{F}$ in degree 0 and 0 otherwise.

Question:

Is this true? More generally, is there a spectral sequence calculating the cohomology of the composite of a right derived- with a left derived functor?

Is there an exact sequence of terms of low degree, as there is for the composite of two right derived functors?