We study the hyperbolicity properties of the action of a non-elementary automorphism group on a compact complex surface, with an emphasis on K3 and Enriques surfaces. A first result is that when such a group contains parabolic elements, Zariski diffuse invariant measures automatically have non-zero Lyapunov exponents. In combination with our previous work, this leads to simple criteria for a uniform expansion property on the whole surface, for groups with and without parabolic elements. This, in turn, has strong consequences on the dynamics: description of orbit closures, equidistribution, ergodicity properties, etc. Along the way, we provide a reference discussion on uniform expansion of non-linear discrete group actions on compact (real) manifolds and the construction of Margulis functions under optimal moment conditions.