  Extragalactic fast X-ray transients (FXTs) are a class of soft (0.3-10 keV)
X-ray transients lasting a few hundred seconds to several hours. Several
progenitor mechanisms have been suggested to produce FXTs, including supernova
shock breakouts, binary neutron star mergers, or tidal disruptions involving an
intermediate-mass black hole and a white dwarf. We present detailed host
studies, including spectroscopic observations of the host galaxies of 7
XMM-Newton-discovered FXTs. The candidate hosts lie at redshifts 0.0928 $< z <$
0.645 implying peak X-ray luminosities of 10$^{43}$ erg s$^{-1}$ $< L_X <$
10$^{45}$ erg s$^{-1}$,and physical offsets of 1 kpc < $r_\mathrm{proj}$ < 22
kpc. These observations increase the number of FXTs with a spectroscopic
redshift measurement by a factor of 2, although we note that one event is
re-identified as a Galactic flare star. We infer host star formation rates and
stellar masses by fitting the combined spectroscopic and archival photometric
data. We also report on a contemporaneous optical counterpart search to the
FXTs in Pan-STARRS and ATLAS by performing forced photometry at the position of
the FXTs. We do not find any counterpart in our search. Given our constraints,
including peak X-ray luminosities, optical limits, and host properties, we find
that XRT 110621 is consistent with a SN SBO event. Spectroscopic redshifts of
likely host galaxies for four events imply peak X-ray luminosities that are too
high to be consistent with SN SBOs, but we are unable to discard either the BNS
or WD-IMBH TDE scenarios for these FXTs.
