In the Milky Way, cosmic rays (CRs) are dynamically important in the interstellar medium, contribute to hydrostatic balance, and may help regulate star formation. However, we know far less about the importance of CRs in galaxies whose gas content or star formation rate differ significantly from those of the Milky Way. Here we construct self-consistent models for hadronic CR transport, losses, and contribution to pressure balance as a function of galaxy properties, covering a broad range of parameters from dwarfs to extreme starbursts. While the CR energy density increases from $sim 1$ eV cm$^{-3}$ to $sim 1$ keV cm$^{-3}$ over the range from sub-Milky Way dwarfs to bright starbursts, strong hadronic losses render CRs increasingly unimportant dynamically as the star formation rate surface density increases. In Milky Way-like systems, CR pressure is typically comparable to turbulent gas and magnetic pressure at the galactic midplane, but the ratio of CR pressure to gas pressure drops to $sim 10^{-3}$ in dense starbursts. Galaxies also become increasingly CR calorimetric and gamma-ray bright in this limit. The degree of calorimetry at fixed galaxy properties is sensitive to the assumed model for CR transport, and in particular to the time CRs spend interacting with neutral ISM, where they undergo strong streaming losses. We also find that in some regimes of parameter space hydrostatic equilibrium discs cannot exist, and in Paper II of this series we use this result to derive a critical surface in the plane of star formation surface density and gas surface density beyond which CRs may drive large-scale galactic winds.
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