  We present VLT/XSHOOTER rest-frame UV-optical spectra of 10 Hot Dust-Obscured
Galaxies (Hot DOGs) at $z\sim2$ to investigate AGN diagnostics and to assess
the presence and effect of ionized gas outflows. Most Hot DOGs in this sample
are narrow-line dominated AGN (type 1.8 or higher), and have higher Balmer
decrements than typical type 2 quasars. Almost all (8/9) sources show evidence
for ionized gas outflows in the form of broad and blueshifted [O III] profiles,
and some sources have such profiles in H$\alpha$ (5/7) or [O II] (3/6).
Combined with the literature, these results support additional sources of
obscuration beyond the simple torus invoked by AGN unification models. Outflow
rates derived from the broad [O III] line ($\rm
\gtrsim10^{3}\,M_{\odot}\,yr^{-1}$) are greater than the black hole accretion
and star formation rates, with feedback efficiencies ($\sim0.1-1\%$) consistent
with negative feedback to the host galaxy's star formation in merger-driven
quasar activity scenarios. We find the broad emission lines in luminous,
obscured quasars are often better explained by outflows within the narrow line
region, and caution that black hole mass estimates for such sources in the
literature may have substantial uncertainty. Regardless, we find lower bounds
on the Eddington ratio for Hot DOGs near unity.
