We present a comprehensive study on the magnetic structure, dynamics, and phase evolution in the single phase double perovskite La2CoMnO6. The mixed valence state due to oxygen deficiency is verified by x-ray photoelectron spectroscopy and confirms a double ferromagnetic transition observed in dc magnetization. Neutron diffraction reveals that the magnetic structure is dominated by long-range ferromagnetic ordering, which is further corroborated by a critical exponents analysis of the paramagnetic to ferromagnetic phase transition. An analysis of the magnetization dynamics by means of linear and nonlinear ac magnetic susceptibilities marks the presence of two distinct cluster-glass-like states that emerge at low temperatures. The isothermal entropy change as a function of temperature and magnetic field (H) is exploited to investigate the mechanism of stabilization of the magnetic phases across the H-T phase diagram. In the regime of the phase diagram where thermal energy is sufficiently low, regions of competing interactions due to local disorder become stabilized and display glasslike dynamics. The freezing mechanism of clusters is illustrated using a unique probe of transverse susceptibility that isolates the effects of the local anisotropy of the spin clusters. The results are summarized in a H-T phase diagram of La2CoMnO6 from these data.