Baryonic Ecosystem IN Galaxies (BEINGMgII) -- III. Cool gas reservoirs at $0.3 \le z \le 1.6$ in the Dark Energy Survey

We investigate the origin of intervening cool MgII absorption detected in the spectra of background quasars and the nature of associated galaxies across a broad redshift range of $0.3 \le z \le 1.6$. Using nebular [O II] $\lambda\lambda$3727,3729 emission lines identified in DESI fiber spectra centered on quasar, we detect 377 galaxies at a typical detection rate of $\sim$0.45% at $z \lesssim 1$, which increases with MgII equivalent width ($W_{2796}$). A significant fraction (74%) of these galaxies are associated with strong absorbers with $W_{2796} \ge$ 2\r{A}. These absorbers trace galaxies spanning stellar masses of $\rm 8.4 \le \log(M_{\star}/M_{\odot}) \le 11.6$ and star formation rates (SFRs) of $\rm -1.2 \le log(SFR~[M_{\odot}yr^{-1}]) \le 2.7$, located at projected galactocentric distances of 4-24 kpc. We find the average MgII absorber strength increases from 2.1\r{A} to 2.9\r{A} between redshifts $z \sim$ 0.4 and 1.2, indicating evolution in the cool gas content of galaxy halos. The relatively constant absorber strength with galactocentric distance implies a clumpy structure of cool gas in the circumgalactic medium (CGM). Further, we find a positive correlation between $W_{2796}$ versus $M_\star$, and SFR, suggesting that the distribution of metal-enriched cool gas in the CGM is closely tied to the properties of the host galaxies. The redshift evolution of gas-phase metallicity suggests that strong MgII absorbers trace the general population of star-forming galaxies. The velocity dispersion of the cool gas increases with halo mass, and the wide range of line of sight velocity offset (-389 to 364 $\rm km\ s^{-1}$) between the galaxy systemic velocity and absorbers highlights the dynamical nature of CGM. However, the majority of this gas remains gravitationally bound to the dark matter halos, consistent with a picture of gas recycling via galactic fountains.