The past decade has seen an explosion of discoveries and new insights into the diffuse gas within galaxies, galaxy clusters, and the filaments composing the Cosmic Web. A new decade will bring fresh opportunities to further this progress towards developing a comprehensive view of the composition, thermal state, and physical processes of diffuse gas in the Universe. Ultraviolet (UV) spectroscopy, probing diffuse 10^4-10^6 K gas at high spectral resolution, is uniquely poised to (1) witness environmental galaxy quenching processes in action, such as strangulation and tidal- and ram-pressure stripping, (2) directly account for the baryon content of galaxy clusters in the cold-warm (T<10^6 K) gas, (3) determine the phase structure and kinematics of gas participating in the equilibrium-regulating exchange of energy at the cores of galaxy clusters, and (4) map cold streams and filaments of the Cosmic Web that feed galaxies and clusters. With a substantial UV undertaking beyond the Hubble Space Telescope, all of the above would be achievable over the entire epoch of galaxy cluster formation. Such capabilities, coupled with already-planned advancements at other wavelengths, will transform extragalactic astronomy by revealing the dominant formation and growth mechanisms of gaseous halos over the mass spectrum, settling the debate between early- and late-time metal enrichment scenarios, and revealing how the ecosystems in which galaxies reside ultimately facilitate their demise.
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