Question about homeomorphic surfaces I am wondering if there exist an example of two surfaces such that they are orientable (or not orientable) and have same Euler characteristic but they are NOT homeomorphic.

Thanks for your support.
Consider a punctured disc or the torus. 

This cannot happen for closed surfaces by the classification theorem of surfaces.
No. By definition, both these surfaces would be homeomorphic to the same connect sum of tori, and so homeomorphic.