  We study the halo mass function and inner halo structure at high redshifts
($z\geq5$) for a suite of simulations within the structure formation ETHOS
framework. Scenarios such as cold dark matter (CDM), thermal warm dark matter
(WDM), and dark acoustic oscillations (DAO) of various strengths are contained
in ETHOS with just two parameters $h_{\rm peak}$ and $k_{\rm peak}$, the
amplitude and scale of the first DAO peak. The Extended Press-Schechter (EPS)
formalism with a smooth-$k$ filter is able to predict the cut-off in the halo
mass function created by the suppression of small scale power in ETHOS models
(controlled by $k_{\rm peak}$), as well as the slope at small masses that is
dependent on $h_{\rm peak}$. Interestingly, we find that DAOs introduce a
localized feature in the mass distribution of haloes, resulting in a mass
function that is distinct in shape compared to either CDM or WDM. We find that
the halo density profiles of ${\it all}$ ETHOS models are well described by the
NFW profile, with a concentration that is lower than in the CDM case in a way
that is regulated by $k_{\rm peak}$. We show that the concentration-mass
relation for DAO models can be well approximated by the mass assembly model
based on the extended Press-Schechter theory, which has been proposed for CDM
and WDM elsewhere. Our results can be used to perform inexpensive calculations
of the halo mass function and concentration-mass relation within the ETHOS
parametrization without the need of $N-$body simulations.
