1. The purpose of choosing a coordinate system other than the obvious rectangular coordinates is to simplify the integral. This is achieved if the region or surface becomes simple in the new coordinate system, and/or if the integrals obtained become doable.
When there is no obvious simplification one can always start with rectangular coordinates.

2. Other coordinates which can be kept in mind are polar (in two dimensions), cylindrical and spherical. Variations in the latter which involve scale changes are called ellipsoidal coordinates. When your region is bounded by certain surfaces, you can conceivably use the functions that define these surfaces as variables.