Using the first-principles exact muffin-tin orbitals method in combination with the coherent potential approximation, we investigated the magnetic properties, exchange interactions, and temperature-dependent half-metallicity of the Co2Mn(Ga1-xZx) (Z=Si, Ge, Sn) alloys. The total magnetic moment follows perfectly a previously proposed Slater-Pauling relation, i.e., μ0 = Nt - 24, with Nt being the number of valence electrons. The Co-Mn and Co1-Co2 (inter-sublattice) interactions are dominated by direct exchange, whereas the Co1-Co1 (intra-sublattice) interaction is characterized by superexchange. The Mn-Mn exchange interaction in Co2MnGa is of long-ranged RKKY-type. However, the Mn-Mn exchange interactions in Co2MnZ are relatively localized and can be attributed to superexchange. The Co-Mn, Co1-Co2 and Co1-Co1 total exchange interactions increase with x, whereas the Mn-Mn total exchange interactions show convex behavior. The calculated Curie temperature (TC) increases with x. The ability of Z to enhance TC follows the sequence of Si > Ge > Sn, in agreement with the experimental findings. The temperature dependence of the spin polarization at the Fermi level [P(T)] is investigated based on the disordered local moment model. P(T) drops abruptly at temperatures much lower than TC. At temperatures higher than 200 K, the composition with higher TC generally corresponds to larger P(T).