  The z-GAL survey observed 137 bright Herschel-selected targets with the IRAM
NOrthern Extended Millimeter Array, with the aim to measure their redshift and
study their properties. Several of them have been resolved into multiple
sources. Consequently, robust spectroscopic redshifts have been measured for
165 individual galaxies in the range 0.8<z<6.5. In this paper we analyse the
millimetre spectra of the z-GAL sources, using both their continuum and line
emission to derive their physical properties. At least two spectral lines are
detected for each source, including transitions of 12CO, [CI], and H2O. The
observed 12CO line ratios and spectral line energy distributions of individual
sources resemble those of local starbursts. In seven sources the
para-H2O(2_11-2_02) transition is detected and follows the IR versus H2O
luminosity relation of sub-millimetre galaxies. The molecular gas mass of the
z-GAL sources is derived from their 12CO, [CI], and sub-millimetre dust
continuum emission. The three tracers lead to consistent results, with the dust
continuum showing the largest scatter when compared to 12CO. The gas-to-dust
mass ratio of these sources was computed by combining the information derived
from 12CO and the dust continuum and has a median value of 107, similar to
star-forming galaxies of near-solar metallicity. The same combined analysis
leads to depletion timescales in the range between 0.1 and 1.0 Gyr, which place
the z-GAL sources between the `main sequence' of star formation and the locus
of starbursts. Finally, we derived a first estimate of stellar masses - modulo
possible gravitational magnification - by inverting known gas scaling
relations: the z-GAL sample is confirmed to be mostly composed by starbursts,
whereas ~25% of its members lie on the main sequence of star-forming galaxies
(within +/- 0.5 dex).
