Collisions of two fermionic dimers near the unitary limit are studied using exact four-particle equations for transition operators in momentum space. Universal properties of dimer-dimer phase shifts and effective range expansion (ERE) parameters are determined. The inclusion of the fourth-order momentum term in the ERE significantly extends its validity to higher collision energies. The dimer-dimer scattering length and effective range are determined in the unitary limit as well as their corrections arising due to the finite range of the two-fermion interaction. These results are of considerably higher accuracy as compared to previous works, but confirm most of the previous results except for the lattice effective field theory calculations.