We present VLT/XSHOOTER rest-frame UV-optical spectra of 10 Hot Dust-Obscured Galaxies (Hot DOGs) at $zsim2$ 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 galaxys 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.
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