We prove disorder universality of chaos phenomena and ultrametricity in the mixed p-spin model under mild moment assumptions on the environment. This establishes the longstanding belief among physicists that the solution of mean-field models with Gaussian disorder also holds for different environments. Our results extend to the mixed p-spin model as well as to different spin glass models. These include universality of quenched disorder chaos in the Edwards-Anderson (EA) model and quenched concentration for the magnetization in both EA and mixed p-spin models under non-Gaussian environments. In addition, we show quenched self-averaging for the overlap in the random field Ising model under small perturbation of the external field.