We obtained comprehensive sets of elemental abundances for eleven damped Ly-alpha systems (DLAs) at z_DLA=1.7-2.5. In Paper I of this series, we showed for three DLA galaxies that we can derive their star formation histories and ages from a detailed comparison of their intrinsic abundance patterns with chemical evolution models. We determine in this paper the star formation properties of six additional DLA galaxies. The derived results confirm that no single star formation history explains the diverse sets of abundance patterns in DLAs. We demonstrate that the various star formation histories reproducing the DLA abundance patterns are typical of local irregular, dwarf starburst and quiescent spiral galaxies. Independent of the star formation history, the DLAs have a common characteristic of being weak star forming galaxies; models with high star formation efficiencies are ruled out. All the derived DLA star formation rates per unit area are moderate or low, with values between -3.2 < log SFR < -1.1 M_sol yr^{-1} kpc^{-2}. The DLA abundance patterns require a large spread in ages ranging from 20 Myr up to 3 Gyr. The oldest DLA in our sample is observed at z_DLA=1.864 with an age estimated to more than 3 Gyr; it nicely indicates that galaxies were already forming at z_f>10. But, most of the DLAs show ages much younger than that of the Universe at the epoch of observation. Young galaxies thus seem to populate the high redshift Universe at z>2, suggesting relatively low redshifts of formation (z~3) for most high-redshift galaxies. The DLA star formation properties are compared with those of other high-redshift galaxies identified in deep imaging surveys with the aim of obtaining a global picture of high-redshift objects.
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