It will be helpful if you look up the definition of a singularity in spacetime in a standard textbook on GR, Hawking/Ellis "The Large Scale Structure of Spacetime" comes to mind.

About your first question:
The important point is that the definition of a singularity is non-trivial, it cannot simply be defined as a point on a Lorentzian manifold where some tensor diverges (despite the name "singularity", the reason for this is explained in most GR textbooks, the Hawking/Ellis book contains a particularly lucid discussion of the involved ideas). Instead, the defining property is that in the presence of a singularity, there are observers aka reference frames that do not live as long as the universe exists, they either end before the universe itself comes to an end (these are the ones who fall into the singularity) or did not exist when the universe was created (these are the ones escaping from the singularity).

The basic idea is therefore to define a spacetime without singularities to be a spacetime that is geodesically complete. Given a spacetime with a singularity, we can remove all geodesics from the spacetime that end up in the singularity and the ones that emerge from the singularity, and get a spacetime without singularities. The minimum region that we have to remove is then defined as the region comprising the singularity. Therefore the "black hole region" or "singularity" of a spacetime is *defined* to be the minimum subset one has to remove to get a "complete" spacetime in the sense that there are no geodesics that end or begin life prematurely.

Page 19 of your reference is a first step in making these ideas a little bit more precise, it is not a mathematical definition, but a heuristic motivation.

Penrose diagrams are a tool in this context, but not a necessary tool.

I don't understand your second question, page 21 seems to be an explanation of the above ideas using generic Penrose diagrams, this is simply a graphical explanation of what the authors have done before, the diagrams are supposed to show a spacetime with a singularity.

As for your third question, I cannot access the Christodoulou paper.