We study a sample of 17 z>1.5 absorbers selected based on the presence of strong CI absorption lines in SDSS spectra and observed with the ESO-VLT spectrograph X-shooter. We derive metallicities, depletion onto dust, and extinction by dust, and analyse the absorption from MgII, MgI, CaII and NaI that are redshifted into the near infrared wavelength range. We show that most of these CI absorbers have high metallicity and dust content. We detect nine CaII absorptions with $W$(CaII$lambda$3934) >0.23 AA out of 14 systems where we have appropriate wavelength coverage. The observed equivalent widths are similar to what has been measured in other lower redshift surveys of CaII systems. We detect ten NaI absorptions in the 11 systems where we could observe this absorption. The median equivalent width ($W$(NaI$lambda$5891) = 0.68 AA) is larger than what is observed in local clouds with similar HI column densities but also in z<0.7 CaII systems detected in the SDSS. The systematic presence of NaI absorption in these CI systems strongly suggests that the gas is neutral and cold, maybe part of the diffuse molecular gas in the ISM of high-redshift galaxies. Most of the systems (12 out of 17) have $W$(MgII$lambda$2796) > 2.5 AA and six of them have log N(HI) < 20.3, with the extreme case of J1341+1852 that has log N(HI) = 18.18. The MgII absorptions are spread over more than $Delta v$ $sim$ 400 km s$^{-1}$ for half of the systems; three absorbers have $Delta v$ > 500 km s$^{-1}$. The kinematics are strongly perturbed for most of these systems, which probably do not arise in quiet disks and must be close to regions with intense star-formation activity and/or are part of interacting objects. All this suggests that a large fraction of the cold gas at high redshift arises in disturbed environments.
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