  Proximity orbital and spin-orbital effects of graphene on monolayer
transition-metal dichalcogenides (TMDCs) are investigated from
first-principles. The Dirac band structure of graphene is found to lie within
the semiconducting gap of TMDCs for sulfides and selenides, while it merges
with the valence band for tellurides. In the former case the proximity-induced
staggered potential gaps and spin-orbit couplings (all on the meV scale) of the
Dirac electrons are established by fitting to a phenomenological effective
Hamiltonian. While graphene on MoS$_2$, MoSe$_2$, and WS$_2$ has a
topologically trivial band structure, graphene on WSe$_2$ exhibits inverted
bands. Using a realistic tight-binding model we find topologically protected
helical edge states for graphene zigzag nanoribbons on WSe$_2$, demonstrating
the quantum spin Hall effect. This model also features "half-topological
states", which are protected against time-reversal disorder on one edge only.
