  This paper provides an update of our previous scaling relations (Genzel et
al.2015) between galaxy integrated molecular gas masses, stellar masses and
star formation rates, in the framework of the star formation main-sequence
(MS), with the main goal to test for possible systematic effects. For this
purpose our new study combines three independent methods of determining
molecular gas masses from CO line fluxes, far-infrared dust spectral energy
distributions, and ~1mm dust photometry, in a large sample of 1444 star forming
galaxies (SFGs) between z=0 and 4. The sample covers the stellar mass range
log(M*/M_solar)=9.0-11.8, and star formation rates relative to that on the MS,
delta_MS=SFR/SFR(MS), from 10^{-1.3} to 10^{2.2}. Our most important finding is
that all data sets, despite the different techniques and analysis methods used,
follow the same scaling trends, once method-to-method zero point offsets are
minimized and uncertainties are properly taken into account. The molecular gas
depletion time t_depl, defined as the ratio of molecular gas mass to star
formation rate, scales as (1+z)^{-0.6}x(delta_MS)^{-0.44}, and is only weakly
dependent on stellar mass. The ratio of molecular-to-stellar mass mu_gas
depends on (1+z)^{2.5}x (delta_MS)^{0.52}x(M*)^{-0.36}, which tracks the
evolution of the specific star formation rate. The redshift dependence of
mu_gas requires a curvature term, as may the mass-dependences of t_depl and
mu_gas. We find no or only weak correlations of t_depl and mu_gas with optical
size R or surface density once one removes the above scalings, but we caution
that optical sizes may not be appropriate for the high gas and dust columns at
high-z.
