In this thesis we describe the generation and analysis of hydrodynamical simulations of galaxy clusters and their intracluster medium (ICM), using large cosmological boxes to generate large samples, in conjunction with individual cluster computations. The main focus is the exploration of the non-thermal processes in the ICM and the effect they have on the interpretation of observations used for cosmological constraints. We provide an introduction to the cosmological structure formation framework for our computations and an overview of the numerical simulations and observations of galaxy clusters. We explore the cluster magnetic field observables through radio relics, extended entities in the ICM characterized by their of diffuse radio emission. We show that statistical quantities such as radio relic luminosity functions and rotation measure power spectra are sensitive to magnetic field models. The spectral index of the radio relic emission provides information on structure formation shocks, e.g., on their Mach number. We develop a coarse grained stochastic model of active galaxy nucleus (AGN) feedback in clusters and show the impact of such inhomogeneous feedback on the thermal pressure profile. We explore variations in the pressure profile as a function of cluster mass, redshift, and radius and provide a constrained fitting function for this profile. We measure the degree of the non-thermal pressure in the gas from internal cluster bulk motions and show it has an impact on the slope and scatter of the Sunyaev-Zel’dovich (SZ) scaling relation. We also find that the gross shape of the ICM, as characterized by scaled moment of inertia tensors, affects the SZ scaling relation. We demonstrate that the shape and the amplitude of the SZ angular power spectrum is sensitive to AGN feedback, and this affects the cosmological parameters determined from high resolution ACT and SPT cosmic microwave background data. We compare analytic, semi-analytic, and simulation-based methods for calculating the SZ power spectrum, and characterize their differences. All the methods must rely, one way or another, on high resolution large-scale hydrodynamical simulations with varying assumptions for modelling the gas of the sort presented here. We show how our results can be used to interpret the latest ACT and SPT power spectrum results. We provide an outlook for the future, describing follow-up work we are undertaking to further advance the theory of cluster science.