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.

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 and Milnor-Thurston 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).