  Various systems of moment equations---consisting of up to (d+3)(d^2+6d+2)/6
moments---in a general dimension d for a dilute granular gas composed of
Maxwell molecules are derived from the inelastic Boltzmann equation by
employing the Grad moment method. The Navier--Stokes-level constitutive
relations for the stress and heat flux appearing in the system of mass,
momentum and energy balance equations are determined from the derived moment
equations. It has been shown that the moment equations only for the
hydrodynamic field variables (density, velocity and granular temperature),
stress and heat flux---along with the time-independent value of the fourth
cumulant---are sufficient for determining the Navier--Stokes-level constitutive
relations in the case of inelastic Maxwell molecules, and that the other
higher-order moment equations do not play any role in this case. The
homogeneous cooling state of a freely cooling granular gas is investigated with
the system of the Grad (d+3)(d^2+6d+2)/6-moment equations and its various
subsystems. By performing a linear stability analysis in the vicinity of the
homogeneous cooling state, the critical system size for the onset of
instability is estimated through the considered Grad moment systems. The
results on critical system size from the presented moment theories are found to
be in reasonably good agreement with those from simulations.
