A systematic study of the two-neutrino double beta (2νββ) decay to the final ground state and excited states is performed within an extension of the quasiparticle random-phase approximation (QRPA) model. In this extension, the multiple commutator model (MCM), a simultaneous treatment of the double-odd and double-even nuclei is possible by assuming their states to have the structure of one or two QRPA phonons. The MCM calculation is done for nine 2 νβ-β- decays and six 2νECEC decays, and systematics of the relevant matrix elements and half-lives have been created. The study of the 2νββ-decay rates is complemented with the MCM study of the single-beta-decay properties of the relevant nuclei within the double-beta isobaric chains. The Woods-Saxon single-particle energies have been corrected near the Fermi surface by comparing the BCS quasiparticle energies with spectroscopic data of odd-mass nuclei. Pairing gaps, energy systematics of the Gamow-Teller states and the available beta-decay data have been used to obtain effective, model-space adapted, two-body matrix elements starting from the G-matrix elements of the Bonn one-boson-exchange potential. This enables a parameter-free calculation of the double Gamow-Teller matrix elements and theoretical prediction of double-beta half-lives.