  We present the first X-ray census of fast radio burst (FRB) host galaxies to
conduct the deepest search for AGN and X-ray counterparts to date. Our sample
includes seven well-localized FRBs with unambiguous host associations and
existing deep Chandra observations, including two events for which we present
new observations. We find evidence for AGN in two FRB host galaxies based on
the presence of X-ray emission coincident with their centers, including the
detection of a luminous ($L_X\approx\,5\times\,10^{42}\,\rm\,erg\,s^{-1}$)
X-ray source at the nucleus of FRB20190608B's host, for which we infer an SMBH
mass of $\rm{M_{BH}\sim\,10^{8}\,M_{\odot}}$ and an Eddington ratio
$\rm{L_{bol}/L_{Edd}\approx\,0.02}$, characteristic of geometrically thin disks
in Seyfert galaxies. We also report nebular emission line fluxes for 24 highly
secure FRB hosts (including 10 hosts for the first time), and assess their
placement on a BPT diagram, finding that FRB hosts trace the underlying galaxy
population. We further find that the hosts of repeating FRBs are not confined
to the star-forming locus, contrary to previous findings. Finally, we place
constraints on associated X-ray counterparts to FRBs in the context of
ultraluminous X-ray sources (ULXs), and find that existing X-ray limits for
FRBs rule out ULXs brighter than $L_X\gtrsim\,10^{40}\,\rm\,erg\,s^{-1}$.
Leveraging the CHIME/FRB catalog and existing ULX catalogs, we search for
spatially coincident ULX-FRB pairs. We identify a total of 28 ULXs spatially
coincident with the localization regions for 17 FRBs, but find that the
DM-inferred redshifts for the FRBs are inconsistent with the ULX redshifts,
disfavoring an association between these specific ULX-FRB pairs.
