The lowest triplet H2O2 potential surface was analyzed for the transition and minimum-energy structures in the range from -0.2 to +5.4 eV with respect to the H-2+O-2 energy. All the transition structures, the reaction pathways, and the local minima were found to have planar configurations for the atoms. We focus on the transition structures responsible for the main bimolecular chemical reactions formally possible on this surface: H-2+O-2<->2HO, H+HO2, and H2O+O; H+HO2<->2HO and H2O+O; and 2HO<->H2O+O. For these reactions, activation energies and rate constants in the transition state approximation were evaluated. Our computed rate constants confirm the recommended values for the H+HO2-->H+O-2 and HO+HO-->H2O+O reactions. The results obtained refute the elementary character of the H+HO2-->H2O+O process and call into question the possibility of chain initiation in the H-2/O-2 system by means of a bimolecular reaction. Most likely, the chain initiation in the gas phase is owing to trimolecular reactions H-2+2O(2)-->2HO(2), 2HO+O-2. Special attention was paid to accurate prediction of electronic energies in the transition structures. A new procedure developed, "extrapolation to zero high-level correction," results in very realistic activation energies. Predictions of molecular energies are coincident with those from the widely used G2 scheme but have smaller uncertainty.