  Graphene/MoS$_2$ van der Waals (vdW) heterostructures have promising
technological applications due to their unique properties and functionalities.
Many experimental and theoretical research groups across the globe have made
outstanding contributions to benchmark the properties of graphene/MoS$_2$
heterostructures. Even though some research groups have already made an attempt
to model the graphene/MoS$_2$ heterostructures using {\it first-principles}
calculations, there exists several discrepancies in the results from different
theoretical research groups and the experimental findings. In the present work,
we revisit this problem by first principles approach and address the existing
discrepancies about the interlayer spacing between graphene and MoS$_2$
monolayers in graphene/MoS$_2$ heterostructures, and the location of Dirac
points near Fermi-level. We find that the Tkatchenko--Scheffler method
efficiently evaluates the long-range vdW interactions and accurately predicts
interlayer spacing between graphene and MoS$_2$ sheets. We further investigate
the electronic, mechanical and vibrational properties of the optimized
graphene/MoS$_2$ heterostructures created using 5$\times$5/4$\times$4 and
4$\times$4/3$\times$3 supercell geometries having different magnitudes of
lattice mismatch. The effect of the varying interlayer spacing on the
electronic properties of heterostructures is discussed. Our phonon calculations
reveal that the interlayer shear and breathing phonon modes, which are very
sensitive to the weak vdW interactions, play vital role in describing the
thermal properties of the studied systems. The thermodynamic and elastic
properties of heterostructures are further discussed. A comparison between our
results and the results reported from other research groups is presented.
