Where to study $2$-category theory? Is there any place where I can read about $2$-categories? I am looking for a proper treatment - there is a section in Borceux's Handbook of Categorical Algebra, but it only sketches some parts of the theory, and it does not include any proofs. If it helps, I would like to get familiar with what is needed in order to be able to deal with the $2$-categorical structures involved in topos theory. 
Is it just me, or does anyone else think that category theory is developing quicker than it can be written down? ... I have the same problem with internal category theory.
Any help would be highly appreciated.
 A: For 2-categories specifically, there is Review of the elements of $2$-categories, by Kelly and Street (1974). Bénabou wrote Introduction to bicategories (1967). 2-category theory is intimately linked to enriched category theory, for which a standard reference is Kelly's Basic concepts of enriched category theory.
Today people also study higher category theory, not just 2-categories1. There's a book by Tom Leinster called Higher Operads, Higher Categories that's very interesting and has the added advantage of also talking about multicategories (also known as colored operads). He does explain in detail what (colored) operads are, but maybe a short introduction to operads would be a good idea before reading it. There's also another book by Cheng and Lauda called Higher-Dimensional Categories: an illustrated guide book that has many helpful pictures.
Another resource is the nLab. It's a wiki not unlike Wikipedia, but devoted to higher category theory.

Regarding your last paragraph: category theory is a very vast field of mathematics, and a very active one at that. A lot of effort was spent trying to make everything precise and clear (as evidenced by the numerous variations on the definition of a higher category, and the many papers, books... devoted to explaining them). In particular there are also the two huge (about a thousand pages each!) books of Jacob Lurie, Higher Topos Theory and Higher Algebra2, who formalizes everything in the form of $(\infty,1)$-categories. As far as I'm aware, it's generally agreed that it's one of the "best" models for higher categories.
1 I've actually learned since then that 2-category theory is still an active field of research on its own! So if you're really interested in 2-categories, go for it!
2 I didn't cite them earlier because I believe it's better to be familiar with higher category theory before tackling them (Lurie himself gives a list of requirements here). But once you're familiar with it, they're are extremely good references
A: I think the combination of Basic Bicategories by Tom Leinster for basic definitions, and A 2-categories companion by Stephen Lack for the essential 2-categorical constructions is a good way to get a feel for the theory, although the latter is a fairly informal and nonrigorous account.
A: 2-categories have been considered for nearly as long as categories. I think it's more that the field isn't large enough to support an exhaustive range of advanced textbooks, and that the natural progression is either to study a specific 2-category or to go on to $n$- and $\infty$-categories. That said, I believe you can find everything you need for toposes in Johnstone's Sketches of an Elephant, and you might also look at Amnon Neeman's monograph on triangulated categories. Kelly and Street, "Review of the Elements of 2-Categories" comes highly recommended, but I've never seen it.
