How does one go about accurately modeling sound propagation in a room (with reflections, absorption, and diffusion characteristics) from the motion of a loud speaker? More specifically what are the governing equations that are needed? Obviously Naiver-Stokes but this is too general? Is there an easier way?
From your description I deduce that approximation of geometrical acoustics should be enough. For its applicability we need to ensure that
The sound could be described as small perturbation (so, no nonlinear effects).
Wavelengths of sound are much smaller than the dimensions of structures with which the sound interacts.
The main equation for geometrical acoustics would be the eikonal equation.
If for your applications the interference effects are essential, then linear acoustics approximation is needed. Main equation would be the wave equation for pressure and/or velocity potential.
Upon assuming small oscillations and neglecting the viscosity of air, the linear three-dimensionsal wave equation with proper boundary conditions would do in the time domain or alternatively Helmholtz equation in the frequency domain. The boundary value problem can be solved by a numerical method e.g. the finite element method.