The diffuse interstellar medium (ISM) plays a pivotal role in the star formation process, but its temperature distribution, turbulent properties, and chemistry are still poorly understood. Recently, the 21-SPONGE survey measured Galactic neutral hydrogen (HI) absorption with exceptional sensitivity, quantifying the cold, warm, and thermally unstable gas (CNM, WNM, and UNM) fractions, the temperature distribution, and the turbulent properties of HI in the direction of 57 bright background sources. For a subset of sources, we have obtained molecular absorption spectra (CCH, HNC, HCN, and HCO+) with ALMA to characterize the chemistry of the molecular ISM along the same lines of sight as 21-SPONGE, where we have key constraints on the kinetic temperature, cold gas fraction, and interstellar turbulence. We have detected molecular gas in the direction of some sources, while some sources show no evidence for molecular gas. Both detections and non-detections are used to place constraints on the role of non-equilibrium chemistry in molecule formation, and to test predictions for molecular abundances based on recent MHD models. We also detect a variety of molecular species in the direction of 3C123, whose line of sight hosts AU-scale overdense and overpressured HI structures. Molecular probes of this AU-scale structure are vital to understand its formation and role in the ISM.