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.