The questions you are asking are fundamental to the theory of one-dimensional dynamical systems.  I would suggest starting with an introductory textbook, such as *An Introduction to Chaotic Dynamical Systems* by Devaney. Books with more in-depth results include *Iterated Maps of the Interval as Dynamical Systems* by Collet and Eckmann, and *One-Dimensional Dynamics* by Melo and van Strien.

By the way, the islands of stability you mention are related to [Sharkovskii's theorem](https://en.wikipedia.org/wiki/Sharkovskii%27s_theorem) and [Milnor-Thurston kneading theory](https://en.wikipedia.org/wiki/Milnor%E2%80%93Thurston_kneading_theory), both of which are covered in Devaney's book.  The self-similarity is a result of something called "renormalization" (which as far as I know is not related to the concept with the same name in quantum field theory).