This work focuses on the development and application of high-order discretization techniques for multi-component turbulent non-equilibrium hypersonic flows. The governing equations (i.e., Navier-Stokes) are discretized in space using finite differences. High-order approximation of the inviscid flux derivatives are sought within the framework of Weighted Essentially Non-Oscillatory (WENO) schemes, with particular emphasis on minimization of dissipation and dispersion errors. Central finite differences are adopted to discretize the diffusive flux derivatives. Time-integration is performed via split/un-split Strong-Stability-Preserving schemes. The proposed numerical methods are implemented in an innovative high-performance tool, hypercode, described in a companion paper. Thermodynamic and transport properties, and source terms due to chemistry are evaluated using the plato library developed at University of Illinois. Applications consider two canonical problems: (i) Taylor-Green vortex and (ii) decay of compressible isotropic turbulence.