  The prospects of extracting new physics signals in coherent elastic
neutrino--nucleus scattering (CE$\nu$NS) processes are limited by the precision
with which the underlying nuclear structure physics, embedded in the weak
nuclear form factor, is known. We present calculations of charge and weak
nuclear form factors and CE$\nu$NS cross sections on $^{12}$C, $^{16}$O,
$^{40}$Ar, $^{56}$Fe and $^{208}$Pb nuclei. We obtain the proton and neutron
densities, and charge and weak form factors by solving Hartree--Fock (HF)
equations with a Skyrme (SkE2) nuclear potential. We validate our approach by
comparing $^{208}$Pb and $^{40}$Ar charge form factor predictions with
available elastic electron scattering data. Since CE$\nu$NS experiments at
stopped--pion sources are also well suited to measure inelastic
charged--current and neutral--current neutrino--nucleus cross sections, we also
present calculations for these processes, incorporating a continuum Random
Phase Approximation (CRPA) description on top of the HF-SkE2 picture of the
nucleus. Providing both coherent as well as inelastic cross sections in a
consistent framework, we aim at obtaining a reliable and detailed comparison of
the strength of these processes in the energy region below ~100 MeV.
Furthermore, we attempt to gauge the level of theoretical uncertainty
pertaining to the description of the $^{40}$Ar form factor and CE$\nu$NS cross
sections by comparing relative differences between recent microscopic nuclear
theory and widely--used phenomenological form factor predictions. Future
precision measurements of CE$\nu$NS will potentially help in constraining these
nuclear structure details that will in turn improve prospects of extracting new
physics.
