We present the general kinetic formalism for the description of spin and energy relaxation of the cavity polaritons in the framework of the Born-Markov approximation. All essential mechanisms of polariton redistribution in reciprocal space together with the final state bosonic stimulation are taken into account from our point of view. The developed theory is applied to describe our experimental results on the polarization dynamics obtained in the polariton parametric amplifier geometry (pumping at the so-called magic angle). Under circular pumping, we show that the spin relaxation time is strongly dependent on the detuning between the exciton and cavity mode energies mainly because of the influence of the detuning on the coupling strength between the photon-like part of the exciton-polariton lower dispersion branch and the reservoir of uncoupled exciton states. In the negative detuning case we find a very long spin relaxation time of about 300 ps. In the case of excitation by a linearly polarized light, we have experimentally confirmed that the anisotropy of the polariton-polariton interaction is responsible for the build-up of the cross-linear polarization of the signal. In the spontaneous regime the polarization degree of the signal is -8% but it can reach -65% in the stimulated regime. The long-living linear polarization observed at zero detuning indicates that the reservoir is formed by excitons localized at the anisotropic islands oriented along the crystallographic axes. Finally, under elliptical pumping, we have directly measured in the time domain and modeled the effect of self-induced Larmor precession, i.e., the rotation of the linear polarization of a state about an effective magnetic field proportional to the projection of the total spin of exciton-polaritons in the cavity on its growth axis.