  Kepler-22b is the first transiting planet to have been detected in the
habitable-zone of its host star. At 2.4 Earth radii, Kepler-22b is too large to
be considered an Earth-analog, but should the planet host a moon large enough
to maintain an atmosphere, then the Kepler-22 system may yet possess a telluric
world. Aside from being within the habitable-zone, the target is attractive due
to the availability of previously measured precise radial velocities and low
intrinsic photometric noise, which has also enabled asteroseismology studies of
the star. For these reasons, Kepler-22b was selected as a target-of-opportunity
by the 'Hunt for Exomoons with Kepler' (HEK) project. In this work, we conduct
a photodynamical search for an exomoon around Kepler-22b leveraging the
transits, radial velocities and asteroseismology plus several new tools
developed by the HEK project to improve exomoon searches. We find no evidence
for an exomoon around the planet and exclude moons of mass >0.5 Earth masses to
95% confidence. By signal injection and blind retrieval, we demonstrate that an
Earth-like moon is easily detected for this planet even when the
time-correlated noise of the data set is taken into account. We provide updated
parameters for the planet Kepler-22b including a revised mass of <53 Earth
masses to 95% confidence and an eccentricity of 0.13(-0.13)(+0.36) by
exploiting Single-body Asterodensity Profiling (SAP). Finally, we show that
Kepler-22b has a >95% probability of being within the empirical habitable-zone
but a <5% probability of being within the conservative habitable-zone.
