We combine James Webb Space Telescope (JWST) and Hubble Space Telescope imaging with Atacama Large Millimeter Array CO(2-1) spectroscopy to study the highly turbulent multiphase intergalactic medium (IGM) in Stephan's Quintet on 25-150 pc scales. Previous Spitzer observations revealed luminous H₂ line cooling across a 45 kpc-long filament, created by a giant shock wave, following the collision with an intruder galaxy, NGC 7318b. We demonstrate that the Mid-Infrared Instrument/F1000W/F770W filters are dominated by 0-0 S(3) H₂ and a combination of polycyclic aromatic hydrocarbon and 0-0 S(5) H₂ emission. These observations reveal the dissipation of kinetic energy as massive clouds experience collisions, interactions, and likely destruction/recycling within different phases of the IGM. In 1 kpc-scaled structure, warm H₂ was seen to form a triangular-shaped head and tail of compressed and stripped gas behind a narrow shell of cold H₂. In another region, two cold molecular clumps with very different velocities are connected by an arrow-shaped stream of warm, probably shocked, H₂ suggesting a cloud-cloud collision is occurring. In both regions, a high warm-to-cold molecular gas fraction indicates that the cold clouds are being disrupted and converted into warm gas. We also map gas associated with an apparently forming dwarf galaxy. We suggest that the primary mechanism for exciting strong mid-IR H₂ lines throughout Stephan's Quintet is through a fog of warm gas created by the shattering of denser cold molecular clouds and mixing/recycling in the post-shocked gas. A full picture of the diverse kinematics and excitation of the warm H₂ will require future JWST mid-IR spectroscopy. The current observations reveal the rich variety of ways that different gas phases can interact with one another.