This paper deals with necessary conditions for minimal solutions of constrained and unconstrained optimization problems with respect to general domination sets by using a well-known nonlinear scalarization functional with uniform level sets (called Gerstewitz' functional in the literature). The primary objective of this work is to establish revised formulas for basic and singular subdifferentials of these nonlinear scalarization functionals and study important properties such as the PSNC property, the Lipschitz behavior, etc. of these scalarization functionals without assuming that the shifted set involved in the definition of the functional is convex. The second objective is to propose a new way to scalarize a set-valued optimization problem. It allows us to study necessary conditions for minimal solutions in a very broad setting in which the domination set is not necessarily convex or solid or conical. The third objective is to apply our results to vector-valued approximation problems.