Your comment about flashbacks and other literary devices that alter the time flow of a narrative leads me to look forward to your group's project. In some past incarnations of this course we have had guest lecturers on that topic, including Prof. Arnold Weinstein who talked about "Light in August" as an example of a story told from several different perspectives. He also liked a novel "L'Emploi du Temps" (="Passing Time") by Michel Butor, a powerful treatment of the space-time experience of a visitor to England trying to keep a dual-time diary while attempting to make sense out of a basically unstructured experience. I hope your group will come up with your own good examples.
Singular points and manifolds are good topics, but hard to deal with on a "synthetic" basis, suppressing the equations that generate the pictures. I definitely think of shock waves emanating from a curve or surface, producing "parallel objects". That is one of the best ways of motivating the study of curvature, one of my favorite concepts.
I agree that the chapter raises questions that are not answered, totally anyway. The idea is that everyone will begin to find such examples in personal experience, and hopefully at least have the vocabulary, and some rudimentary tools, for dealing with them.