  We study the metallicity dependence of the H/H$_2$ and C$^+$/C/CO
distributions in a self-regulated interstellar medium (ISM) across a broad
range of metallicities ($0.1 < Z/Z_\odot < 3$). To this end, we conduct
high-resolution (particle mass of $1\ {\rm M_\odot}$) hydrodynamical
simulations coupled with a time-dependent H$_2$ chemistry network. The results
are then post-processed with an accurate chemistry network to model the
associated C$^+$/C/CO abundances, based on the time-dependent non-steady-state
(``non-equilibrium'') H$_2$ abundances. We find that the time-averaged star
formation rate and the ISM structure are insensitive to metallicity. The column
densities relevant for molecular shielding appear correlated with the volume
densities in gravitationally unstable gas. As metallicity decreases, H$_2$
progressively deviates from steady state (``equilibrium'') and shows shallow
abundance profiles until they sharply truncate at the photodissociation fronts.
In contrast, the CO profile is sharp and controlled by photodissociation as CO
quickly reaches steady state. We construct effective one-dimensional cloud
models that successfully capture the time-averaged chemical distributions in
simulations. At low metallicities, the steady-state model significantly
overestimates the abundance of H$_2$ in the diffuse medium. The overestimated
H$_2$, however, has little impact on CO. Consequently, the mass fraction of
CO-dark H$_2$ gas is significantly lower than what a fully steady-state model
predicts. The mass ratios of H$_2$/C$^+$ and H$_2$/C both show a weaker
dependence on $Z^{\prime}$ than H$_2$/CO, which potentially indicates that
C$^+$ and C could be alternative tracers for H$_2$ at low $Z^{\prime}$ in terms
of mass budget. Our chemistry code for post-processing is publicly available.
