Hybrids of graphene and two-dimensional transition-metal dichalcogenides (TMDCs) have the potential to bring graphene spintronics to the next level. As we show here by performing first-principles calculations of graphene on monolayer MoS2, there are several advantages of such hybrids over pristine graphene. First, Dirac electrons in graphene exhibit a giant global proximity spin-orbit coupling, without compromising the semimetallic character of the whole system at zero field. Remarkably, these spin-orbit effects can be very accurately described by a simple effective Hamiltonian. Second, the Fermi level can be tuned by a transverse electric field to cross the MoS2 conduction band, creating a system of coupled massive and massless electron gases. Both charge and spin transport in such systems should be unique. Finally, we propose to use graphene/TMDC structures as a platform for optospintronics, in particular, for optical spin injection into graphene and for studying spin transfer between TMDCs and graphene.