We present a simultaneous calculation for the shape evolution and the related spectroscopic properties of the low-lying states, and the β-decay properties in the even-and odd-mass Ge and As nuclei in the mass A ≈ 70 − 80 region, within the framework of the nuclear density functional theory and the particle-core coupling scheme. The constrained self-consistent mean-field calculations using a universal energy density functional (EDF) and a pairing interaction determines the interactingboson Hamiltonian for the even-even core nuclei, and the essential ingredients of the particle-boson interactions for the odd-nucleon systems, and of the Gamow-Teller and Fermi transition operators. A rapid structural evolution from γ-soft oblate to prolate shapes, as well as the spherical-oblate shape coexistence around the neutron sub-shell closure N = 40, is suggested to occur in the eveneven Ge nuclei. The predicted low-energy spectra, electromagnetic transition rates, and β-decay log f t values are in a reasonable agreement with experiment. The predicted log f t values reflect the structures of the wave functions for the initial and final nuclei of β decay, which are to a large extent determined by the microscopic input provided by the underlying EDF calculation.