  The WISE mission has unveiled a rare population of high-redshift ($z=1-4.6$),
dusty, hyper-luminous galaxies, with infrared luminosities $L_{\rm IR} >
10^{13}~L_{\odot}$, and sometimes exceeding $10^{14}~L_{\odot}$. Previous work
has shown that their dust temperatures and overall far-IR SEDs are
significantly hotter than expected for star-formation. We present here an
analysis of the rest-frame optical through mid-IR SEDs for a large sample of
these so-called "Hot, Dust-Obscured Galaxies" (Hot DOGs). We find that the SEDs
of Hot DOGs are generally well modeled by the combination of a luminous, yet
obscured AGN that dominates the rest-frame emission at $\lambda > 1\mu\rm m$
and the bolometric luminosity output, and a less luminous host galaxy that is
responsible for the bulk of the rest optical/UV emission. Even though the
stellar mass of the host galaxies may be as large as
$10^{11}-10^{12}~M_{\odot}$, the AGN emission, with luminosities comparable to
those of the most luminous QSOs known, require that either Hot DOGs have black
hole masses significantly in excess of the local relations, or that they
radiate significantly above the Eddington limit. We show that, while rare, the
number density of Hot DOGs is comparable to that of equally luminous but
unobscured (i.e., Type 1) QSOs. This is inconsistent with the trend of a
diminishing fraction of obscured objects with increasing luminosity found for
less luminous QSOs, possibly indicating a reversal in this relation at high
luminosity, and that Hot DOGs are not the torus-obscured counterparts of the
known optically selected, largely unobscured Hyper-Luminous QSOs. Hot DOGs may
represent a different type of galaxy and thus a new component of the galaxy
evolution paradigm. Finally, we discuss the environments of Hot DOGs and show
that these objects are in regions as dense as those of known high-redshift
proto-clusters.(Abridged)
