We use single-dish radio spectra of known 22 GHz H$_2$O megamasers, primarily gathered from the large dataset observed by the Megamaser Cosmology Project, to identify Keplerian accretion disks and to investigate several aspects of the disk physics. We test a mechanism for maser excitation proposed by Maoz & McKee (1998), whereby population inversion arises in gas behind spiral shocks traveling through the disk. Though the flux of redshifted features is larger on average than that of blueshifted features, in support of the model, the high-velocity features show none of the predicted systematic velocity drifts. We find rapid intra-day variability in the maser spectrum of ESO 558-G009 that is likely the result of interstellar scintillation, for which we favor a nearby ($D approx 70$ pc) scattering screen. In a search for reverberation in six well-sampled sources, we find that any radially-propagating signal must be contributing $lesssim$10% of the total variability. We also set limits on the magnetic field strengths in seven sources, using strong flaring events to check for the presence of Zeeman splitting. These limits are typically 200--300 mG ($1sigma$), but our most stringent limits reach down to 73 mG for the galaxy NGC 1194.
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