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.