  We investigate the electronic dispersion and transport properties of
graphene/WSe$_{2}$ heterostructures in the presence of a proximity induced
spin-orbit coupling (SOC) using a low-energy Hamiltonian, with different types
of symmetry breaking terms, obtained from a four-band, first and second
nearest-neighbour tight-binding (TB) one. The competition between different
perturbation terms leads to inverted SOC bands. Further, we study the effect of
symmetry breaking terms on ac and dc transport by evaluating the corresponding
conductivities within linear response theory. The scattering-independent part
of the valley-Hall conductivity, as a function of the Fermi energy $E_{F}$, is
mostly negative in the ranges $-\lambda_{R}\leqslant E_{F}$ and
$E_{F}\geqslant\lambda_{R}$ when the strength $\lambda_{R}$ of the Rashba SOC
increases except for a very narrow region around $E_{F}=0$ in which it peaks
sharply upward. The scattering-dependent diffusive conductivity increases
linearly with electron density, is directly proportional to $\lambda_{R}$ in
the low- and high-density regimes, but weakens for $\lambda_{R}=0$. We
investigate the optical response in the presence of a SOC-tunable band gap for
variable $E_{F}$. An interesting feature of this SOC tuning is that it can be
used to switch on and off the Drude-type intraband response. Furthermore, the
ac conductivity exhibits interband responses due to the Rashba SOC. We also
show that the valley-Hall conductivity changes sign when $E_F$ is comparable to
$\lambda_R$ and vanishes at higher values of $E_F$. It also exhibits a strong
dependence on temperature and a considerable structure as a function of the
frequency.
