The regulation of the interstellar medium (ISM) by stellar feedback is a fundamental process in galaxy evolution, governing the thermal balance, chemical enrichment, and dynamical state of baryonic matter. This dissertation presents a comprehensive spatially resolved X-ray spectroscopy as a case study of the diffuse hot ISM in the nearby grand-design spiral galaxy M51 (NGC 5194), with the aim of constraining the physical conditions of the feedback-heated plasma and their spatial dependence. Deep Chandra observations are analyzed using both conventional annular spectral fitting and a newly developed hierarchical Bayesian framework that jointly models thousands of low–signal-to-noise spectra. The hierarchical approach allows for simultaneous inference of local and population-level parameters—including temperature, emission measure, and intrinsic absorption—while rigorously propagating statistical uncertainties and mitigating biases arising from spatial averaging. The results reveal that the X-ray–emitting plasma is dominated by components with temperatures $\leq$ 0.1 keV, whose broadband emission is heavily attenuated. Emission measures are systematically enhanced along spiral arms relative to interarm regions, primarily reflecting higher density and/or filling factors, whereas metallicities show no significant arm–interarm variation. Comparisons with high-resolution hydrodynamical simulations indicate broad agreement in large-scale thermal structures but also identify discrepancies in the prevalence of low-temperature gas, with implications for feedback efficiency and energy coupling. This work advances the application of hierarchical Bayesian inference in X-ray astronomy, establishes a framework for statistically robust spectral mapping of diffuse emission, and provides quantitative constraints essential for testing and refining theoretical models of stellar feedback in star-forming galaxies.