Using Green's-function many-body theory, we present the details of a formally exact adiabatic-connection fluctuation-dissipation density-functional theory based on range separation, which was sketched in Toulouse et al. [Phys. Rev. Lett. 102, 096404 (2009)]. Range-separated density-functional theory approaches combining short-range density-functional approximations with long-range random-phase approximations (RPAs) are then obtained as well-identified approximations on the long-range Green's-function self-energy. Range-separated RPA-type schemes with or without long-range Hartree-Fock exchange response kernel are assessed on rare gas and alkaline-earth-metal dimers and compared to range-separated second-order perturbation theory and range-separated coupled-cluster theory.