We report Atacama Large Millimeter Array observations of the neutral atomic carbon transitions [C i] and multiple CO lines in a sample of ̃30 main-sequence galaxies at $z\sim 1$ , including novel information on [C i] ${(}^{3}{P}_{2}{\mbox{--}}^{3}{P}_{1})$ and CO $(7\mbox{--}6)$ for 7 of such normal objects. We complement our observations with a collection of >200 galaxies with coverage of similar transitions, spanning the z = 0─4 redshift interval and a variety of ambient conditions from local to high-redshift starbursts. We find systematic variations in the [C i]/IR and [C i]/high-Jupper (Jupper = 7) CO luminosity ratios among the various samples. We interpret these differences as increased dense molecular gas fractions and star formation efficiencies in the strongest high-redshift starbursts with respect to normal main-sequence galaxies. We further report constant ${L}_{{[{\rm{C}}{\rm{I}}]}^{3}{P}_{2}\,-{}^{3}{P}_{1}}^{{\prime} }$ / ${L}_{{[{\rm{C}}{\rm{I}}]}^{3}{P}_{1}\,-{}^{3}{P}_{0}}^{{\prime} }$ ratios across the galaxy populations and redshifts, suggesting that gas temperatures Texc traced by [C i] do not strongly vary. We find only a mild correlation with Tdust and that, generally, Texc ≲ Tdust. We fit the line ratios with classical photodissociation region models, retrieving consistently larger densities and intensities of the UV radiation fields in submillimeter galaxies than in main-sequence and local objects. However, these simple models fall short in representing the complexity of a multiphase interstellar medium and should be treated with caution. Finally, we compare our observations with the Santa Cruz semi-analytical model of galaxy evolution, recently extended to simulate submillimeter emission. While we confirm the success in reproducing the CO lines, we find systematically larger [C i] luminosities at fixed IR luminosity than predicted theoretically. This highlights the necessity of improving our understanding of the mechanisms regulating the [C i] emission on galactic scales. We release our data compilation to the community.