TITLE: Derived categories of (coherent) sheaves of modules: exceptional images, gluing, and proper descent?
QUESTION [12 upvotes]: I am interested in the properties of (the derived categories) of various categories of (coherent) sheaves of modules (over varieties). I would like to understand in what extent these properties are similar to those of (etale) constructible sheaves and what are the differences. I have looked at Derived categories of coherent sheaves: suggested references?
but I still do not know the answers to the following questions:
1. Do the categories in question become 'very bad' if the base variety is not proper or is not smooth?
2. Are there (derived) exceptional images (i.e. $Rf^!$ and $Rf_!$) defined in this setting?
3. Could one 'glue' somehow (some version) of the derived category of sheaves of modules over $X$ from those over $Z$ and over $X\setminus Z$ ($Z$ is a closed subvariety of $X$)? 
4. Does there exist a version of proper descent for this setting?
Any comments or references would be very welcome!

REPLY [5 votes]: 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!