GW170817, the first detection of gravitational waves (GWs) from a binary neutron star merger together with its electromagnetic counterpart has proven the immense potential of this multi-messenger astronomy for our understanding of the nature of gravity and the properties of dense matter. We propose here to advance on the latter, addressing the following questions: To which extent can current and future GW observations constrain the underlying microphysics, specifically the equation of state and neutrino interaction rates? Can a potential phase transition be detected? To answer these questions we will study as well the remnant of binary neutron star mergers as proto-neutron stars, the compact postbounce objects in a core-collapse supernova with a new fast and flexible numerical tool. We shall largely explore the microphysics model's parameter space and simulate the possible imprints on GW emission from these objects. In view of the difficult GW data analysis, this work will be crucial to fully exploit the science of upcoming GW data.