What exactly is a Yang-Mills Theory? Is it a general theory, based on SU(N) symmetry, which can then be applied for particular cases (ElectroWeak, Chromodynamics) ? Is it like a general mathematical model of gauge theory ?
Not necessarily SU(N). It can also be like SO(N), Sp(N), etc. The idea is we need to introduce gauge fields to keep the theory gauge invariant.
The gauge group of the field theory needs to be nonabelian, else it would just be quantum electrodynamics.