Kapustin-Orlov'a survey of derived categories of coherent sheaves is pretty good,
- A. N. Kapustin, D. O. Orlov, Lectures on mirror symmetry, derived categories, and D-branes, Uspehi Mat. Nauk 59 (2004), no. 5(359), 101--134; translation in Russian Math. Surveys 59 (2004), no. 5, 907--940, math.AG/0308173
but more slow/elementary exposition starting with fundamentals of derived categories is in an earlier survey of Orlov
- D. O. Orlov, Derived categories of coherent sheaves and equivalences between them, Uspekhi Mat. Nauk, 2003, Vol. 58, issue 3(351), pp. 89–172, Russian pdf, English transl. in Russian Mathematical Surveys (2003),58(3):511, doi link, pdf at Orlov's webpage (not on arXiv!)
There are also Orlov's handwritten slides in djvu from a 5-lecture course in Bonn
- djvu, but the link is temporary
For derived categories per se, apart from Gelfand-Manin methods book and Weibel's homological algebra remember that a really good expositor is Bernhard Keller. E.g. his text
- Bernhard Keller, Introduction to abelian and derived categories, pdf
...and also his Handbook of Algebra entry on derived categories:
pdf
1. It depends what you mean by bad. The categories still do what they are meant to do even if the underlying variety is not proper or smooth. However, there are some subtleties. For instance, if you try to pushforward a coherent sheaf along a non-proper morphism, of course the result might only be quasicoherent. Likewise, if you try to pullback a coherent sheaf from a non-regular base, then the result might be unbounded (since you might have to take an infinitely long locally free resolution). However, perfect complexes (an intrinsically defined subcategory of $D^{b}_{Coh}(X)$ behave well under pullback.
2. Since you already have problems for pullback and pushforward, $Rf^!$ and $Rf_!$ could also not exist in non-proper or non-smooth settings. However, under fairly general hypotheses these functors will exist for the unbounded derived category of quasi-coherent sheaves. See for instance the Springer Lecture Notes of Lipman on Grothendieck duality and references therein.
3. Gluing (recollement) can be a bit of a problem depending on the codimension of $Z$. I think it is fine for $Z$ of codimension at least $2$. (Actually, see the below comment of t3suji pointing out that codimension doesn't help.) You always have a localization sequence
$$D^{b}_{Z, Coh(X)}(X) \rightarrow D^{b}_{Coh(X)}(X) \rightarrow D^{b}(X \setminus Z).$$
However, $D^{b}_{Z, Coh(X)}(X) \rightarrow D^{b}_{Coh(X)}(X)$ might not have the desired adjoints to get a recollement and so glue. The right adjoint would be local cohomology, but these might not be coherent. ( However, you do have gluing for unbounded derived categories of quasi-coherent sheaves.
4. I don't know the answer to this, but probably reading Toen would help. I would guess that you have to work not with triangulated categories but with their natural dg enhancements, since morphisms don't glue in the triangulated categories. That is, already for the identity morphism, I think you have problems. For instance, take a short exact sequence of vector bundles that doesn't split, so the connecting homomorphism is non-trivial. However, the sequence splits locally, so the connecting homomorphism is locally trivial.
Best Answer
I think that a good example of the usefulness of the Derived Category of coherent sheaves for studying classical questions is the recent preprint by Soheyla Feyzbakhsh
Mukai's program (reconstructing a K3 surface from a curve) via wall-crossing,
where the author uses wall-crossing with respect to Bridgeland stability conditions in order to solve the classical Mukai problem of reconstructing a K3 surface from its hyperplane section.