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\textbf{Purpose:} This paper addresses long-standing solar physics problems, namely, the heating of the solar chromosphere and the origin of the solar wind. Our aim is to reveal the related mechanisms behind chromospheric heating and plasma outflows in a quiet-Sun. \textbf{Methods:} The approach is based on a two-fluid numerical model that accounts for thermal non-equilibrium (ionization/recombination), non-adiabatic, and non-ideal dynamics of protons+electrons and hydrogen atoms. The model is applied to numerically simulate the propagation and dissipation of granulation-generated waves in the chromosphere and plasma flows inside a quiet region. \textbf{Results:} The obtained results demonstrate that collisions between protons+electrons and hydrogen atoms supplemented by plasma viscosity, magnetic resistivity, and recombination lead to thermal energy release, which compensates radiative and thermal losses in the chromosphere, and sustains the atmosphere with vertical profiles of averaged temperature and periods of generated waves that are consistent with recent observational data. \textbf{Conclusion:} Our model conjectures a most robust and global physical picture of granulation-generated wave motions, plasma flows, and subsequent heating, which form and dynamically couple the various layers of the solar atmosphere.