We present first-principles calculations of the electronic structure and spin texture of a MoS$_{2}$ monolayer in Fe/MgO/MoS$_{2}$ multilayers. These metal/insulator/semiconductor stackings are subject to an electron transfer from the Fe layer to the MoS$_{2}$ sheet, giving rise to a two-dimensional (2D) electron gas, the density of which depends on the MgO layer thickness. We describe the consequences of this electron transfer and of the magnetic proximity effect on the occupation of the conduction bands of the MoS$_{2}$ layer, on the nature of its band gap, and on the splitting and dispersion of its valence bands near the Γ point of the 2D Brillouin zone. The spin splitting and spin texture are reproduced and understood by an effective Hamiltonian, which includes Rashba, Dresselhaus, and Zeeman effects. We finally show that the splitting of the MoS$_{2}$ valence bands induced near Γ by the spin-orbit coupling is rather different when Fe is replaced by a nonmagnetic transition metal such as vanadium.