Recently fabricated InSe monolayers exhibit remarkable characteristics that indicate the potential of this material to host a number of many-body phenomena. Here, we consistently describe collective electronic effects in hole-doped InSe monolayers using advanced many-body techniques. To this end, we derive a realistic electronicstructure model from first principles that takes into account the most important characteristics of this material, including a flat band with prominent van Hove singularities in the electronic spectrum, strong electron-phonon coupling, and weakly-screened long-ranged Coulomb interactions. We calculate the temperature-dependent phase diagram as a function of band filling and observe that this system is in a regime with coexisting charge density wave and ferromagnetic instabilities that are driven by strong electronic Coulomb correlations. This regime can be achieved at realistic doping levels and high enough temperatures, and can be verified experimentally. We find that the electron-phonon interaction does not play a crucial role in these effects, effectively suppressing the local Coulomb interaction without changing the qualitative physical picture.