  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.
