  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]
