The goal of the project is to establish and analyze infinite-volume Gibbs point processes in random environment, where the environment enters via the intensity measure of the reference Poisson point process. More precisely, we investigate Cox-Gibbs point processes both in the quenched and the annealed sense via GNZ and DLR equations and derive criteria for the quasilocality of the associated Papangelou intensities and specification kernels. Moreover, we feature situations in which the random environment has a significant impact on the quasilocality properties of the model as well as on the phase-transition behavior. In particular, we intend to derive new phase transitions for the uniqueness of the infinite-volume equilibrium system based on connectivity properties and the dimension of the environment. As paradigmatic examples we investigate the Widom-Rowlinson model in an environment of Poisson-Boolean as well as Poisson-Voronoi type.