  We report constraints on a variety of non-standard cosmological models using
the full 5-year photometrically-classified type Ia supernova sample from the
Dark Energy Survey (DES-SN5YR). Both Akaike Information Criterion (AIC) and
Suspiciousness calculations find no strong evidence for or against any of the
non-standard models we explore. When combined with external probes, the AIC and
Suspiciousness agree that 11 of the 15 models are moderately preferred over
Flat-$\Lambda$CDM suggesting additional flexibility in our cosmological models
may be required beyond the cosmological constant. We also provide a detailed
discussion of all cosmological assumptions that appear in the DES supernova
cosmology analyses, evaluate their impact, and provide guidance on using the
DES Hubble diagram to test non-standard models. An approximate cosmological
model, used to perform bias corrections to the data holds the biggest potential
for harbouring cosmological assumptions. We show that even if the approximate
cosmological model is constructed with a matter density shifted by
$\Delta\Omega_m\sim0.2$ from the true matter density of a simulated data set
the bias that arises is sub-dominant to statistical uncertainties.
Nevertheless, we present and validate a methodology to reduce this bias.
