Let $f: X \to Y$ be a proper birational morphism of Noetherian quasi-separated schemes. We have the derived pull-back $Lf^*: D(QCoh(Y))\to D(QCoh(X))$ (https://stacks.math.columbia.edu/tag/06YI) and similarly the derived push-forward $Rf_*: D(QCoh(X))\to D(QCoh(Y))$ functor. I have the following questions:
(1) Is $Rf_*$ a right-adjoint to $Lf^*$ ?
(2) Is $Lf^* \mathcal O_Y\cong O_X$ ?
(3) Does $Rf_*$ map $D(Coh(X))$ to $D(Coh(Y))$ and $D^b(Coh(X))$ to $D^b(Coh(Y))$ ?
(4) Does $Lf^*$ map $D(Coh(Y))$ to $D(Coh(X))$ and $D^b(Coh(Y))$ to $D^b(Coh(X))$ ?
Perhaps all these are standard and well-known, in which case, I would highly appreciate some references as well. Thank you.