We analyze the interstellar medium properties of a sample of sixteen bright CO line emitting galaxies identified in the ALMA Spectroscopic Survey in the Hubble Ultra Deep Field (ASPECS) Large Program. This CO$-$selected galaxy sample is complemented by a couple of additional CO line emitters in the UDF that are identified based on their MUSE optical spectroscopic redshifts. The ASPECS CO$-$selected galaxies cover a larger range of star-formation rates and stellar masses compared to literature CO emitting galaxies at $z>1$ for which scaling relations have been established previously. Most of ASPECS CO-selected galaxies follow these established relations in terms of gas depletion timescales and gas fractions as a function of redshift, as well as the star-formation rate-stellar mass relation (`galaxy main sequence'). However, we find that $\sim30\%$ of the galaxies (5 out of 16) are offset from the galaxy main sequence at their respective redshift, with $\sim12\%$ (2 out of 16) falling below this relationship. Some CO-rich galaxies exhibit low star-formation rates, and yet show substantial molecular gas reservoirs, yielding long gas depletion timescales. Capitalizing on the well-defined cosmic volume probed by our observations, we measure the contribution of galaxies above, below, and on the galaxy main sequence to the total cosmic molecular gas density at different lookback times. We conclude that main sequence galaxies are the largest contributor to the molecular gas density at any redshift probed by our observations (z$\sim$1$-$3). The respective contribution by starburst galaxies above the main sequence decreases from z$\sim$2.5 to z$\sim$1, whereas we find tentative evidence for an increased contribution to the cosmic molecular gas density from the passive galaxies below the main sequence....