We study a two-dimensional, off-lattice particle model introduced to describe absorbing phase transitions in driven non-Brownian suspensions. We numerically explore the (φ, ϵ) phase diagram, where φ is the packing fraction and ϵ controls the amplitude of particle jumps. We use a binary mixture to suppress crystallization, which allows us to disentangle the model’s distinct phase transitions between amorphous states. At large φ, we find that the approach to the absorbing transition is preceded by a non-equilibrium glass transition to a non-diffusive amorphous state. This dynamic arrest makes the location of the critical absorbing transition protocol-dependent. The ϵ → 0 end-point of the transition line defines a jamming transition whose location is shown to vary continuously with the preparation protocol, and cannot serve as a unique definition of random close packing. Near jamming, we observe a complex landscape and marginal stability, reminiscent of Gardner phases found in thermal glasses. The critical exponents characterizing packings at the jamming transition numerically agree with alternative approaches based on energy minimization, and with analytic predictions from mean-field replica theory. We analyze hyperuniformity of large-scale density fluctuations in fluid and glass phases, where it emerges with qualitatively distinct signatures, and show that random organization dynamics does not determine the hyperuniformity observed in jammed packings, which is found to be non-universal. Our results show that random organization models share deep physical similarities with thermal soft-particle systems undergoing glass and jamming transitions, with little impact of the non-equilibrium nature of the microscopic dynamics on emerging physical properties.