  We report on a Hubble Space Telescope search for rest-frame ultraviolet
emission from the host galaxies of five far-infrared-luminous $z\simeq{}6$
quasars and the $z=5.85$ hot-dust free quasar SDSS J0005-0006. We perform 2D
surface brightness modeling for each quasar using a Markov-Chain Monte-Carlo
estimator, to simultaneously fit and subtract the quasar point source in order
to constrain the underlying host galaxy emission. We measure upper limits for
the quasar host galaxies of $m_J>22.7$ mag and $m_H>22.4$ mag, corresponding to
stellar masses of $M_\ast<2\times10^{11}M_\odot$. These stellar mass limits are
consistent with the local $M_{\textrm{BH}}$-$M_\ast$ relation. Our flux limits
are consistent with those predicted for the UV stellar populations of
$z\simeq6$ host galaxies, but likely in the presence of significant dust
($\langle A_{\mathrm{UV}}\rangle\simeq 2.6$ mag). We also detect a total of up
to 9 potential $z\simeq6$ quasar companion galaxies surrounding five of the six
quasars, separated from the quasars by 1.4''-3.2'', or 8.4-19.4 kpc, which may
be interacting with the quasar hosts. These nearby companion galaxies have UV
absolute magnitudes of -22.1 to -19.9 mag, and UV spectral slopes $\beta$ of
-2.0 to -0.2, consistent with luminous star-forming galaxies at $z\simeq6$.
These results suggest that the quasars are in dense environments typical of
luminous $z\simeq6$ galaxies. However, we cannot rule out the possibility that
some of these companions are foreground interlopers. Infrared observations with
the James Webb Space Telescope will be needed to detect the $z\simeq6$ quasar
host galaxies and better constrain their stellar mass and dust content.
