  Monolayer MoS$_2$ has emerged as an interesting material for nanoelectronic
and optoelectronic devices. The effect of substrate screening and defects on
the electronic structure of MoS$_2$ are important considerations in the design
of such devices. Here, we present ab initio density functional theory (DFT) and
GW calculations to study the effect of substrate screening on the quasiparticle
band gap and defect charge transition levels (CTLs) in monolayer MoS$_2$. We
find a giant renormalization to the free-standing quasiparticle band gap by 350
meV and 530 meV in the presence of graphene and graphite as substrates,
respectively. Our results are corroborated by recent experimental measurements
on these systems using scanning tunneling spectroscopy and photoluminescence
excitation spectroscopy. Sulfur vacancies are the most abundant native defects
found in MoS$_2$. We study the CTLs of these vacancies in MoS$_2$ using the
DFT+GW formalism. We find (+1/0) and (0/-1) CTLs appear in the pristine band
gap of MoS$_2$. Substrate screening results in renormalization of the (0/-1)
level, with respect to the valence band maximum (VBM), by the same amount as
the gap. This results in the pinning of the (0/-1) level about $\sim$500 meV
below the conduction band minimum for the free-standing case as well as in the
presence of substrates. The (+1/0) level, on the other hand, lies less than 100
meV above the VBM for all the cases.
