It is also used in eliptic and hyperelliptic curve cryptography, when you study the Jacobian Variety a curve: http://modular.math.washington.edu/edu/Fall2003/252/lectures...

Categories and Functors come out naturally when studying such structures.

I think category theory is rarely essentially used in a first or second typical course on some topic in math. Once in a while if students can be assumed to have had a course in category theory, then it is more elegant and efficient e.g. simply to show that some functor has an adjoint hence this or that limit is preserved, but to me category theory, except if it is the main topic of interest, is useful because it eases communication and allows you to quickly get some understanding of a construction that might look very "local" to some category that you are not familiar with. Unlike the person who created the cited notes, I think it actually can help understanding the things under study or the associated constructions, assuming sufficient command of category theory. I remember when I first understood the definitions of things like products, coproducts, push-outs and pull- backs etc. in my last year as an undergraduate. Suddenly, for many of the constructions from topology and algebra it became easier to remember them, how they were constructed and which properties they had. To me it is kind of like with design patterns, mainly I don't use them as tools picked up from my tool box when solving a problem, rather they simply allow me to communicate more easily and gives me another level of abstraction where I can reuse thinking I have done earlier.

A professor is a "high-ranking" teacher at a university. You can have a professor in pretty much any field, so there are math professors.