We present results from spectroscopic observations with X-shooter at the Very Large Telescope of seven H2-bearing DLAs at high redshifts (z$_{rm abs}sim 2.5-3$). These DLAs were originally selected from the presence of strong H$_2$ lines directly seen at the DLA redshift in low-resolution, low S/N SDSS spectra. We confirm the detection of molecular hydrogen in all of them. We measure the column densities of HI, H$_2$ in various rotational levels, and metal species, and associated dust extinction. The metallicities, obtained from undepleted species, are in the range logZ=-0.8 to -0.2. We discuss the chemical enrichment in these clouds and compare their properties with that of other molecular-rich systems selected by other means. In particular, we show that three different methods of pre-selection of H$_2$-bearing DLAs in the SDSS have their own biases but complement each other mostly in terms of chemical enrichment. We use the rotational excitation of H$_2$ molecules together with the fine-structure energy levels of neutral carbon to constrain the physical conditions in the gas with the help of numerical modeling as well as analytical expressions for the surface density at which atomic to molecular conversion happens. We find that the H$_2$-bearing medium revealed by the studied DLAs has typical values for the kinetic temperature, hydrogen density, and UV radiation field of, respectively, T$sim$100K, $n_{rm H}sim 100$ cm$^{-3}$, and I$_{rm UV}$ about twice the intensity of the Draine field. Detailed studies combining different selections should, therefore, bring important clues to understand the HI-H$_2$ transition at high redshift.
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