The orbits of Saturn's inner mid-sized moons (Mimas, Enceladus, Tethys, Dione, and Rhea) have been notably difficult to reconcile with their geology. Here, we present numerical simulations coupling thermal, geophysical, and simplified orbital evolution for 4.5 billion years that reproduce observed characteristics of their orbits and interiors, provided that the outer four moons are old. Tidal dissipation within Saturn expands the moons' orbits over time. Dissipation within the moons decreases their eccentricities, which are episodically increased by moon-moon interactions, causing past or present oceans in the interior of Enceladus, Dione, and Tethys. In contrast, Mimas' proximity to Saturn's rings generates interactions that cause such rapid orbital expansion that Mimas must have formed only 0.1-1 Gyr ago if it postdates the rings. The resulting lack of radionuclides keeps it geologically inactive. These simulations can explain the Mimas-Enceladus dichotomy, reconcile the moons' orbital properties and geological diversity, and self-consistently produce a recent ocean on Enceladus.