We study a sample of 17 z > 1.5 absorbers selected based on the presence of strong C I absorption lines in Sloan Digital Sky Survey (SDSS) spectra and observed with the European Southern Observatory Very Large Telescope (ESO-VLT) spectrograph X-shooter. We derive metallicities, depletion onto dust, and extinction by dust, and analyse the absorption from Mg II, Mg I, Ca II, and Na I that are redshifted into the near infrared wavelength range. We show that most of these C I absorbers have high metallicity and dust content. We detect nine Ca II absorptions with W(Ca IIλ3934) > 0.23 Å 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 Ca II systems. We detect ten Na I absorptions in the 11 systems where we could observe this absorption. The median equivalent width (W(Na Iλ5891) = 0.68 Å) is larger than what is observed in local clouds with similar H I column densities but also in z < 0.7 Ca II systems detected in the SDSS. The systematic presence of Na I absorption in these C I systems strongly suggests that the gas is neutral and cold, maybe part of the diffuse molecular gas in the interstellar medium of high-redshift galaxies. Most of the systems (12 out of 17) have W(Mg IIλ2796) > 2.5 Å and six of them have log N(H I) < 20.3, with the extreme case of J1341+1852 that has log N(H I) = 18.18. The Mg II absorptions are spread over more than Δυ ~ 400 km s⁻¹ for half of the systems; three absorbers have Δυ > 500 km s⁻¹. The kinematics are strongly perturbed for most of these systems, which indicates that these systems 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.