  Making the most of the rapidly increasing population of gravitational-wave
detections of black hole (BH) and neutron star (NS) mergers requires comparing
observations with population synthesis predictions. In this work we investigate
the combined impact from the key uncertainties in population synthesis
modelling of the isolated binary evolution channel: the physical processes in
massive binary-star evolution and the star formation history as a function of
metallicity, $Z$, and redshift $z, \mathcal{S}(Z,z)$. Considering these
uncertainties we create 560 different publicly available model realizations and
calculate the rate and distribution characteristics of detectable BHBH, BHNS,
and NSNS mergers. We find that our stellar evolution and $\mathcal{S}(Z,z)$
variations can impact the predicted intrinsic and detectable merger rates by
factors $10^2$-$10^4$. We find that BHBH rates are dominantly impacted by
$\mathcal{S}(Z,z)$ variations, NSNS rates by stellar evolution variations and
BHNS rates by both. We then consider the combined impact from all uncertainties
considered in this work on the detectable mass distribution shapes (chirp mass,
individual masses and mass ratio). We find that the BHNS mass distributions are
predominantly impacted by massive binary-star evolution changes. For BHBH and
NSNS we find that both uncertainties are important. We also find that the shape
of the delay time and birth metallicity distributions are typically dominated
by the choice of $\mathcal{S}(Z,z)$ for BHBH, BHNS and NSNS. We identify
several examples of robust features in the mass distributions predicted by all
560 models, such that we expect more than 95% of BHBH detections to contain a
BH $\gtrsim 8\,\rm{M}_{\odot}$ and have mass ratios $\lesssim 4$. Our work
demonstrates that it is essential to consider a wide range of allowed models to
study double compact object merger rates and properties.
