  We measure homogeneous distances to M31 and 38 associated stellar systems
($-$16.8$\le M_V \le$ $-$6.0), using time-series observations of RR Lyrae stars
taken as part of the Hubble Space Telescope Treasury Survey of M31 Satellites.
From $>700$ orbits of new/archival ACS imaging, we identify $>4700$ RR Lyrae
stars and determine their periods and mean magnitudes to a typical precision of
0.01 days and 0.04 mag. Based on Period-Wesenheit-Metallicity relationships
consistent with the Gaia eDR3 distance scale, we uniformly measure heliocentric
and M31-centric distances to a typical precision of $\sim20$ kpc (3%) and
$\sim10$ kpc (8%), respectively. We revise the 3D structure of the M31 galactic
ecosystem and: (i) confirm a highly anisotropic spatial distribution such that
$\sim80$% of M31's satellites reside on the near side of M31; this feature is
not easily explained by observational effects; (ii) affirm the thin (rms $7-23$
kpc) planar "arc" of satellites that comprises roughly half (15) of the
galaxies within 300 kpc from M31; (iii) reassess physical proximity of notable
associations such as the NGC 147/185 pair and M33/AND XXII; and (iv) illustrate
challenges in tip-of-the-red-giant branch distances for galaxies with $M_V >
-9.5$, which can be biased by up to 35%. We emphasize the importance of RR
Lyrae for accurate distances to faint galaxies that should be discovered by
upcoming facilities (e.g., Rubin Observatory). We provide updated luminosities
and sizes for our sample. Our distances will serve as the basis for future
investigation of the star formation and orbital histories of the entire known
M31 satellite system.
