  We present rest-frame optical spectroscopic observations of 24 Hot
Dust-Obscured Galaxies (Hot DOGs) at redshifts 1.7-4.6 with KECK/NIRES. Our
targets are selected based on their extreme red colors to be the highest
luminosity sources from the WISE infrared survey. In 20 sources with
well-detected emission we fit the key [O III], H$\beta$, H$\alpha$, [N II], and
[S II] diagnostic lines to constrain physical conditions. Of the 17 targets
with a clear detection of the [O III]$\rm \lambda$5007A emission line, 15
display broad blueshifted and asymmetric line profiles, with widths ranging
from 1000 to 8000 $\rm km\ s^{-1}$ and blueshifts up to 3000 $\rm km\ s^{-1}$.
These kinematics provide strong evidence for the presence of massive ionized
outflows of up to $8000\ \rm M_\odot\ yr^{-1}$, with a median of $150\ \rm
M_\odot\ yr^{-1}$. As many as eight sources show optical emission line ratios
consistent with vigorous star formation. Balmer line star-formation rates,
uncorrected for reddening, range from 30--1300 $\rm M_\odot\ yr^{-1}$, with a
median of $50\ \rm M_\odot\ yr^{-1}$. Estimates of the SFR from SED fitting of
mid and far-infrared photometry suggest significantly higher values. We
estimate the central black hole masses to be of order $10^{8-10}\rm\ M_\odot$,
assuming the present-day $\rm M_{BH}-\sigma_*$ relation. The bolometric
luminosities and the estimated masses of the central black holes of these
galaxies suggest that many of the AGN-dominated Hot DOGs are accreting at or
above their Eddington limit. The combination of ongoing star formation, massive
outflows, and high Eddington ratios suggest Hot DOGs are a transitional phase
in galaxy evolution.
