We report full quantum scattering calculations of inelastic cross sections and rate coefficients for Mg in collision with H atoms. Transitions until the Mg(3s3d) 3D level were considered, as well as mutual neutralization and ion-pair production. These calculations are based on new ab initio potentials and nonadiabatic couplings for the nine and five lowest electronic states of MgH. The collision energy range is from threshold up to 10 eV. The main mechanisms underlying the scattering processes are discussed. An extension of these calculations is performed by means of a multichannel model approach. Rate coefficients are thus obtained for all processes between the 12 lowest atomic states of Mg in collisions with H, plus the ionic channel. Those data permit to model spectral lines in stellar atmospheres. Their impact on statistical equilibrium of Mg populations through the non-LTE departure coefficients is investigated for the Sun and the HD122563 metal-poor star.