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!

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!