  We present observations of CO(3-2) in 13 main-sequence $z=2.0-2.5$
star-forming galaxies at $\log(M_*/M_{\odot})=10.2-10.6$ that span a wide range
in metallicity (O/H) based on rest-optical spectroscopy. We find that
CO(3-2)/SFR decreases with decreasing metallicity, implying that the CO
luminosity per unit gas mass is lower in low-metallicity galaxies at $z\sim2$.
We constrain the CO-to-H$_2$ conversion factor ($\alpha_{\text{CO}}$) and find
that $\alpha_{\text{CO}}$ inversely correlates with metallicity at $z\sim2$. We
derive molecular gas masses ($M_{\text{mol}}$) and characterize the relations
among $M_*$, SFR, $M_{\text{mol}}$, and metallicity. At $z\sim2$,
$M_{\text{mol}}$ increases and molecular gas fraction ($M_{\text{mol}}$/$M_*$)
decrease with increasing $M_*$, with a significant secondary dependence on SFR.
Galaxies at $z\sim2$ lie on a near-linear molecular KS law that is
well-described by a constant depletion time of 700 Myr. We find that the
scatter about the mean SFR-$M_*$, O/H-$M_*$, and $M_{\text{mol}}$-$M_*$
relations is correlated such that, at fixed $M_*$, $z\sim2$ galaxies with
larger $M_{\text{mol}}$ have higher SFR and lower O/H. We thus confirm the
existence of a fundamental metallicity relation at $z\sim2$ where O/H is
inversely correlated with both SFR and $M_{\text{mol}}$ at fixed $M_*$. These
results suggest that the scatter of the $z\sim2$ star-forming main sequence,
mass-metallicity relation, and $M_{\text{mol}}$-$M_*$ relation are primarily
driven by stochastic variations in gas inflow rates. We place constraints on
the mass loading of galactic outflows and perform a metal budget analysis,
finding that massive $z\sim2$ star-forming galaxies retain only 30% of metals
produced, implying that a large mass of metals resides in the circumgalactic
medium.
