Our current understanding of the Universe rests on two pillars: General Relativity, which governs its large-scale evolution, and the Standard Model of particle physics, which describes the behaviour and properties of its constituent particles. Over the last few decades however, observational tensions with these two models have emerged. Notably, they do not provide a mechanism to explain the measured neutrino masses or the origins of both baryonic and dark matter. All of these observations strongly suggest the existence of beyond-the-Standard-Model (BSM) physics. Over the years, a plethora of such BSM scenarios have been developed to explain various combinations of these observations. In order to discriminate between these scenarios and constrain new physics, it is therefore of critical importance to improve the accuracy of theoretical predictions for various key observables and leverage the possible interplay between the broad range of cosmological and laboratory experiments searching for such new physics. In this thesis, we apply tools from nonequilibrium Quantum Field Theory, best suited to describe these systems, to improve the theoretical treatment of three key cosmological processes: 1) the thermal production of cosmological gravitational wave backgrounds, 2) the production of the matter-antimatter asymmetry of our Universe and its connection to the neutrino mass generation mechanism, as well as 3) the decoupling of neutrinos from the rest of the primordial plasma. In addition, we also explore various phenomenological consequences of these processes. Even though these three processes take place in widely separated cosmological eras, we show that thermal effects can significantly affect each of them. We also illustrate how the interplay between various cosmological and laboratory experiments can greatly reduce the viable parameter space for new physics.