  The formation redshift and abundance of the first stars and galaxies is
highly sensitive to the build up of low mass dark matter halos as well as
astrophysical feedback effects which modulate star formation in these low mass
halos. The 21-cm signal at cosmic dawn will depend strongly on the formation of
these first luminous sources and thus can be used to constrain unknown
astrophysical and dark matter properties in the early universe. In this paper,
we explore how well we could measure properties of dark matter using the 21-cm
power spectrum at $z>10$, given unconstrained astrophysical parameters. We
create a generalizable form of the dark matter halo mass function for models
with damped and/or oscillatory linear power spectra, finding a single
"smooth-k" window function which describes a broad range of models including
CDM. We use this to make forecasts for structure formation using the Effective
Theory of Structure Formation (ETHOS) framework to explore a broad parameter
space of dark matter models. We make predictions for the 21-cm power spectrum
observed by HERA varying both cosmological ETHOS parameters as well as
astrophysical parameters. Using a Markov Chain Monte Carlo forecast we find
that the ETHOS dark matter parameters are degenerate with astrophysical
parameters linked to star formation in low mass dark matter halos but not with
X-ray heating produced by the first generation of stars. After marginalizing
over uncertainties in astrophysical parameters we demonstrate that with just
540 days of HERA observations it should be possible to distinguish between CDM
and a broad range of dark matter models with suppression at wavenumbers
$k\lesssim 200\,h$Mpc$^{-1}$ assuming a moderate noise level. These results
demonstrate the potential of 21-cm observations to constrain the matter power
spectrum on scales smaller than current probes.
