  We present detailed optical, near-infrared, and radio observations of the
X-ray flash 050416a obtained with Palomar and Siding Springs Observatories as
well as HST and the VLA, placing this event among the best-studied X-ray
flashes to date. In addition, we present an optical spectrum from Keck LRIS
from which we measure the redshift of the burst, z=0.6528. At this redshift the
isotropic-equivalent prompt energy release was about 10^51 erg, and using a
standard afterglow synchrotron model we find that the blastwave kinetic energy
is a factor of 10 larger, E_K,iso ~ 10^52 erg. The lack of an observed jet
break to t ~ 20 days indicates that the opening angle is larger than 7 deg and
the total beaming-corrected relativistic energy is larger than 10^50 erg. We
further show that the burst produced a strong radio flare at t ~ 40 days
accompanied by an observed flattening in the X-ray band which we attribute to
an abrupt circumburst density jump or an episode of energy injection (either
from a refreshed shock or off-axis ejecta). Late-time observations with HST
show evidence for an associated supernova with peak optical luminosity roughly
comparable to that of SN 1998bw. Next, we show that the host galaxy of XRF
050416a is actively forming stars at a rate of at least 2 M_solar per year with
a luminosity of L_B ~ 0.5L* and metallicity of Z ~ 0.2-0.8 Z_solar. Finally, we
discuss the nature of XRF 050416a in the context of short-hard gamma-ray bursts
and under the framework of off-axis and dirty fireball models for X-ray
flashes.
