  The Boltzmann kinetic theory for a model of a confined quasi-two dimensional
granular mixture derived previously [Garz\'o, Brito and Soto, Phys. Fluids
\textbf{33}, 023310 (2021)] is considered further to analyze two different
problems. First, a linear stability analysis of the hydrodynamic equations with
respect to the homogeneous steady state (HSS) is carried out to identify the
conditions for stability as functions of the wave vector, the coefficients of
restitution, and the parameters of the mixture. The analysis, which is based on
the results obtained by solving the Boltzmann equation by means of the
Chapman--Enskog method to first order in spatial gradients, takes into account
the (nonlinear) dependence of the transport coefficients and the cooling rate
on the coefficients of restitution and applies in principle to arbitrary values
of the concentration, and the mass and diameter ratios. In contrast to the
results obtained in the conventional inelastic hard sphere (IHS) model, the
results show that all the hydrodynamic modes are stable so that, the HSS is
linearly \emph{stable} with respect to long enough wavelength excitations. As a
second application, segregation induced by both a thermal gradient and gravity
is studied. A segregation criterion based on the dependence of the thermal
diffusion factor $\Lambda$ on the parameter space of the mixture is derived.
Comparison with previous results derived from the IHS model is carried out.
