Many processes within galaxy clusters, such as those believed to govern the onset of thermally unstable cooling and active galactic nucleus feedback, are dependent upon local dynamical timescales. However, accurate mapping of the mass distribution within individual clusters is challenging, particularly toward cluster centers where the total mass budget has substantial radially dependent contributions from the stellar (M-*), gas (M-gas), and dark matter (M-DM) components. In this paper we use a small sample of galaxy clusters with deep Chandra observations and good ancillary tracers of their gravitating mass at both large and small radii to develop a method for determining mass profiles that span a wide radial range and extend down into the central galaxy. We also consider potential observational pitfalls in understanding cooling in hot cluster atmospheres, and find tentative evidence for a relationship between the radial extent of cooling X-ray gas and nebular Ha emission in cool-core clusters. At large radii the entropy profiles of our clusters agree with the baseline power law of K alpha r(1.1) expected from gravity alone. At smaller radii our entropy profiles become shallower but continue with a power law of the form K alpha r(0.67) down to our resolution limit. Among this small sample of cool-core clusters we therefore find no support for the existence of a central flat "entropy floor."