Interfacing superconductors with strongly spin-polarized magnetic materials opens up the possibility to discover new and enhanced types of low-temperature spintronics devices where spin-triplet Cooper pairs play a key role. Motivated by the recent derivation of spin-polarized quasiclassical boundary conditions capable of describing such a scenario in the diffusive limit, we here consider the emergent physics in diffusive hybrid structures comprised of a conventional s-wave superconductor (e.g. Nb, Al) and either strongly spin-polarized ferromagnetic insulators (e.g. EuO, GdN) or half-metallic ferromagnets (e.g. CrO2, LCMO). Unlike the majority of previous works, we here focus on how the superconductor itself is influenced by the proximity effect, and how the generated triplet Cooper pairs manifest themselves in the self-consistently computed density of states (DOS) and the superconducting critical
temperature Tc of the device. We provide a comprehensive treatment of how the superconductor and its properties are affected by the triplet pairs. We show that our theory can reproduce the recent observation of an unusually large zero-energy peak in a superconductor interfaced with a half-metal, which even exceeds the normal-state DOS. Our results also shed light on the recent experimental observation of a large Tc change in superconductor/half-metal structures, and suggests that there may be other physics at work besides a long-ranged triplet proximity effect.