We investigate systematically the dynamical mass ejection, r-process nucleosynthesis, and properties of electromagnetic counterparts of neutron-star (NS) mergers in dependence on the uncertain properties of the nuclear equation of state (EoS) by employing 40 representative, microphysical high-density EoSs in relativistic, hydrodynamical simulations. The crucial parameter determining the ejecta mass is the radius R_1.35 of a 1.35 M_sun NS. NSs with smaller R_1.35 (soft EoS) eject systematically higher masses. These range from ~10^-3 M_sun to ~10^-2 M_sun for 1.35-1.35 M_sun binaries and from ~5*10^-3 M_sun to ~2*10^-2 M_sun for 1.2-1.5 M_sun systems (with kinetic energies between ~5*10^49 erg and 10^51 erg). Correspondingly, the bolometric peak luminosities of the optical transients of symmetric (asymmetric) mergers vary between 3*10^41 erg/s and 14*10^41 erg/s (9*10^41 erg/s and 14.5*10^41 erg/s) on timescales between ~2 h and ~12 h. If these signals with absolute bolometric magnitudes from -15.0 to -16.7 are measured, the tight correlation of their properties with those of the merging NSs might provide valuable constraints on the high-density EoS. The r-process nucleosynthesis exhibits a remarkable robustness independent of the EoS, producing a nearly solar abundance pattern above mass number 130. By the r-process content of the Galaxy and the average production per event the Galactic merger rate is limited to 4*10^-5/yr (4*10^-4/yr) for a soft (stiff) NS EoS, if NS mergers are the main source of heavy r-nuclei. The production ratio of radioactive 232Th to 238U attains a stable value of 1.64-1.67, which does not exclude NS mergers as potential sources of heavy r-material in the most metal-poor stars.
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