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Proximate molecular quasar absorbers (PH2) are an intriguing population of absorption systems recently uncovered through strong H2 absorption at small velocity separation from the background quasars. We performed a multi-wavelength spectroscopic follow-up of thirteen such systems with VLT/X-Shooter. Here, we present the observations and study the overall chemical enrichment measured from the HI, H2 and metal lines. We combine this with an investigation of the neutral gas kinematics with respect to the quasar host. We find gas-phase metallicities in the range 2% to 40% of the Solar value, i.e. in the upper-half range of HI-selected proximate damped Lyman-alpha systems, but similar to what is seen in intervening H2-bearing systems. This is likely driven by similar selection effects that play against the detection of most metal and molecular rich systems in absorption. Differences are however seen in the abundance of dust (from [Zn/Fe]) and its depletion pattern, when compared to intervening systems, possibly indicating different dust production or destruction close to the AGN. We also note the almost-ubiquitous presence of a high-ionisation phase traced by NV in proximate systems. In spite of the hard UV field from the quasars, we found no strong overall deficit of neutral argon, at least when compared to intervening DLAs. This likely results from argon being mostly neutral in the H2 phase, which actually accounts for a large fraction of the total amount of metals. We measure the quasar systemic redshifts through emission lines from both ionised gas and CO(3-2) emission, the latter being detected in all 6 cases for which we obtained 3-mm data from complementary NOEMA observations. For the first time, we observe a trend between line-of-sight velocity with respect to systemic redshift and metallicity of the absorbing gas. [truncated]