This is just a slight addition to previous answers of Chris and Anatoly.

There are some very beautiful, more-recent-than-the-classical-references papers of Neeman and Lipman-Neeman on (roughly) the subject of question 2. Warning though: in the first (Neeman only) paper, $f^!$ means the right adjoint of $\mathbb{R}f_*$, which won't agree with the $f^!$ defined e.g. in Deligne's appendix to Hartshorne's "Residues and Duality" without a properness hypothesis. [By the way, this appendix is a classical reference.]

Also, as in Anatoly's answer, you shouldn't ask any questions in this area at the abelian level. :-) And indeed (as Anatoly indicates too), without some additional hypotheses (for example quasicompact and quasiseparated) you will have to be more careful about which derived category you want to work in.

This is just a slight addition to previous answers of Chris and Anatoly.

There are some very beautiful, more-recent-than-the-classical-references papers of Neeman and Lipman-Neeman on (roughly) the subject of question 2. Warning though: in the first (Neeman only) paper, $f^!$ means the right adjoint of $\mathbb{R}f_*$, which won't agree with the $f^!$ defined e.g. in Deligne's appendix to Hartshorne's "Residues and Duality" without a properness hypothesis. [By the way, this appendix is a classical reference.]

Also, as in Anatoly's answer, you shouldn't ask any questions in this area at the abelian level. :-) And indeed, without some additional hypotheses you will have to be more careful about which derived category you want to work in. EDIT: If you are interested mainly in a right adjoint to $\mathbb{R}f_*$ then quasicompact quasiseparated will do; this is the setting of Neeman/Lipman-Neeman. Thanks to Anatoly's help, though, I have finally noticed that the original post doesn't seem to indicate caring about this, so perhaps the observation is a useless one to make!