You are asking too many questions, some of them are very difficult.
Here are some answers. 

1. 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).

2. 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.

3. 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.

4. 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.

5. 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.

6. One can give a complete topological description of pre-images under rational
functions and under polynomials, if this is what you want.

Please, be more specific.