  In this work, we resolve conflicting experimental and theoretical findings
related to the dynamical stability and superconducting properties of
$Fm\overline{3}m$-LuH$_3$, which was recently suggested as the parent phase
harboring room-temperature superconductivity at near-ambient pressures.
Including temperature and quantum anharmonic lattice effects in our
calculations, we demonstrate that the theoretically predicted structural
instability of the $Fm\overline{3}m$ phase near ambient pressures is suppressed
for temperatures above $200\,\text{K}$. We provide a $p\,\unicode{x2013}\,T$
phase diagram for stability up to pressures of $6\,\text{GPa}$, where the
required temperature for stability is reduced to $T>80\,\text{K}$. We also
determine the superconducting critical temperature $T_\text{c}$ of
$Fm\overline{3}m$-LuH$_3$ within the Migdal-Eliashberg formalism, using
temperature- and quantum-anharmonically-corrected phonon dispersions, finding
that the expected $T_\text{c}$ for electron-phonon mediated superconductivity
is in the range of $50$ $\unicode{x2013}$ $60\,\text{K}$, i.e., well below the
temperatures required to stabilize the lattice. When considering moderate
doping based on rigidly shifting the Fermi level, $T_\text{c}$ decreases for
both hole and electron doping. Our results thus provide evidence that any
observed room-temperature superconductivity in pure or doped
$Fm\overline{3}m$-LuH$_3$, if confirmed, cannot be explained by a conventional
electron-phonon mediated pairing mechanism.
