Combining 66 ultraviolet (UV) spectra and ancillary data from the recent Low-Redshift Lyman Continuum Survey (LzLCS) and 23 LyC observations by earlier studies, we form a statistical sample of star-forming galaxies at z~0.2-0.4 with which we study the role of cold interstellar medium (ISM) gas in the leakage of ionizing radiation. We also aim to establish empirical relations between the HI neutral and low-ionization state (LIS) absorption lines with different galaxy properties. We first constrain the massive star content (stellar ages and metallicities) and UV attenuation by fitting the stellar continuum with a combination of simple stellar population models. The models, together with accurate LyC flux measurements, allow us to determine the absolute LyC photon escape fraction for each galaxy (f_esc_^abs^f_esc_^abs^). We then measure the equivalent widths and residual fluxes of multiple HI and LIS lines, and the geometrical covering fraction of the UV emission, adopting the picket-fence model. The LyC escape fraction spans a wide range, with a median f_esc_^abs^f_esc_^abs^ (0.16, 0.84 quantiles) of 0.04 (0.02, 0.20), and 50 out of the 89 galaxies detected in the LyC (1{sigma} upper limits of f_esc_^abs^<=0.01 for non-detections, typically). The HI and LIS line equivalent widths scale with the UV luminosity and attenuation, and inversely with the residual flux of these lines. Additionally, Ly{alpha} equivalent widths scale with both the HI and LIS residual fluxes, but anti-correlate with the corresponding HI or LIS equivalent widths. The HI and LIS residual fluxes are correlated, indicating that the neutral gas is spatially traced by the low-ionization transitions. We find that the observed trends of the absorption lines and the UV attenuation are primarily driven by the geometric covering fraction of the gas. The observed nonuniform gas coverage also demonstrates that LyC photons escape through low-column-density channels in theISM. The equivalent widths and residual fluxes of both the HI and LIS lines strongly correlate with f_esc_^abs^ : strong LyC leakers (highest f_esc_^abs^) show weak absorption lines, low UV attenuation, and large Ly{alpha} equivalent widths. We provide several empirical calibrations to estimate f_esc_^abs^ from UV absorption lines. Finally, we show that simultaneous UV absorption line and dust attenuation measurements can, in general, predict the escape fraction of galaxies. We apply our method to available measurements of UV LIS lines of 15 star-forming galaxies at z~4-6 (plus 3 high-z galaxy composites), finding that these high-redshift, UV-bright galaxies (M_UV_<=-21) may have low escape fractions, f_esc_^abs^<=0.1. UV absorption lines trace the coldISM gas of galaxies, which governs the physics of the LyC escape. We show that, with some assumptions, the absolute LyC escape can be statistically predicted using UV absorption lines, and the method can be applied to study galaxies across a wide redshift range, including in the epoch of cosmic reionization.