If the main features of the core-collapse supernova mechanism are now understood, some important details about the microphysics are still subject to a lot of uncertainties. In particular, neutrinos are playing a key role in the core-collapse mechanism but interactions rates between neutrinos and dense matter are still poorly understood. In this talk, I will start by reviewing what we actually know about core-collapse supernovae and neutron star formation by focusing on the role of neutrinos. Then I will present a code for proto-neutron star evolution and some studies we have been conducting on the influence of neutrino - dense matter interactions on core-collapse and (proto-)neutron star cooling.