I understand your situation, but you will find that to learn functional programming you will need to adjust your point of view to the documentation you find, instead of the other way around. Luckily in Scala you have the possibility of becoming a functional programmer gradually.
To answer your questions and explain the point-of-view difference, I need to distinguish between "type classes" (monoids, functors, arrows), mathematically called "structures", and generic operations or algorithms (catamorphisms or folds, anamorphisms or unfolds, etc.). These two often interact, since many generic operations are defined for specific classes of data types.
You look for prescriptive answers similar to design patterns: when does this concept apply? The truth is that you have surely seen the prescriptive answers, and they are simply the definitions of the different concepts. The problem (for you) is that those answers are intrinsically different from design patterns, but it is so for good reasons.
On the one hand, generic algorithms are not design patterns, which suggest a structure for the code you write; they are abstractions defined in the language which you can directly apply. They are general descriptions for common algorithms which you already implement today, but by hand. For instance, whenever you are computing the maximum element of a list by scanning it, you are hardcoding a fold; when you sum elements, you are doing the same; and so on. When you recognize that, you can declare the essence of the operation you are performing by calling the appropriate fold function. This way, you save code and bugs (no opportunity for off-by-one errors), and you save the reader the effort to read all the needed code.
On the other hand, structures concern not the goal you have in mind but properties of the entities you are modeling. They are more useful for bottom-up software construction, rather than top-down: when defining your data, you can declare that it is a e.g. a monoid. Later, when processing your data, you have the opportunity to use operations on e.g. monoids to implement your processing. In some cases it is useful to strive to express your algorithm in terms of the predefined ones. For instance, very often if you need to reduce a tree to a single value, a fold can do most or all of what you need. Of course, you can also declare that your data type is a monoid when you need a generic algorithm on monoids; but the earlier you notice that, the earlier you can start reusing generic algorithms for monoids.
Last advice is that probably most of the documentation you will find about these concepts concerns Haskell, because this language has been around for much more time and supports them in a quite elegant way. Quite recommended here are Learn you a Haskell for Great Good, a Haskell course for beginners, where among others chapters 11 to 14 focus on some type classes, and Typeclassopedia (which contains links to various articles with specific examples). EDIT: Finally, an example of applications of Monoids, taken from Typeclassopedia, is here: http://apfelmus.nfshost.com/articles/monoid-fingertree.html. I'm not saying there is little documentation for Scala, just that there is more in Haskell, and Haskell is where the application of these concepts to programming was born.