  Many compelling dark matter (DM) scenarios feature Coulomb-like interactions
between DM particles and baryons, in which the cross section for elastic
scattering scales with relative particle velocity as $v^{-4}$. Previous studies
have invoked such interactions to produce heat exchange between cold DM and
baryons and alter the temperature evolution of hydrogen. In this study, we
present a comprehensive study of the effects of Coulomb-like scattering on
structure formation, in addition to the known effects on the thermal history of
hydrogen. We find that interactions which significantly alter the temperature
of hydrogen at Cosmic Dawn also dramatically suppress the formation of galaxies
that source the Lyman-$\alpha$ background, further affecting the global 21-cm
signal. In particular, an interaction cross section at the current
observational upper limit leads to a decrease in the abundance of star-forming
halos by a factor of $\sim 2$ at $z\sim 20$, relative to cold, collisionless
DM. We also find that DM that is 100% millicharged cannot reproduce the depth
and the timing of the reported EDGES anomaly in any part of the parameter
space. These results critically inform modeling of the global 21-cm signal and
structure formation in cosmologies with DM-baryon scattering, with
repercussions for future and upcoming cosmological data analysis.
