You are asking too many questions, some of them are very difficult. Here are some answers.
Image of the Jordan curve under a rational function. Take a circle for $\gamma$. Every continuous function on the circle can be uniformly approximated by a finite trigonometric sum (=Laurent polynomial). Laurent polynomial is a rational function. So the image can be "any" closed curve you want (must be algebraic, of course, but can uniformly approximate any closed curve).
For polynomials/entire functions this is not so. Because we have argument principle: The index with respect to any point in the complement of the curve must be zero for a point far away, and non-negative for every other point. So for example, the image cannot have the shape of figure 8. However, a complete topological characterization of images under polynomials is a difficult topological problem.
The difference between polynomials and entire functions you cannot tell (there is no difference in the topology of the image) because polynomials are dense in the space of entire functions.
Then you ask about pre-images. Notice that pre-image can be disconnected. Components of preimage may be non-Jordan: they are ramified at the critical points of your function.
Components of pre-image of a Jordan region under an entire functions are simply connected (this follows from maximum princile). Components of preimage under an entire function can be unbounded.
One can give a complete topological description of pre-images under rational functions and under polynomials, if this is what you want.
EDIT. For example, consider a preimage of a Jordan curve under a polynomial. Take any finite forest (union of disjoint trees) embedded in the plane. Replace every vertex by a topological disk, and let the disks touch wherever there is an edge. This is a topological model of the preimage. If you want to fix the degree of the polinomial, then additionally you prescribe to each disk a positive integer (the degree of the ramified covering), and sum of these integers is the degree of your polynomial.
For rational functions the description is similar but more complicated.
For entire functions it is still more complicated because now we not only deal with an infinite forest, but the "discs" can be unbounded, and quite complicated ( a disk can have uncountable set of accessible points at infinity, etc.)