Stripped-envelope stars can be observed as Wolf-Rayet (WR) stars or as less luminous hydrogen-poor stars with low mass-loss rates and transparent winds. Both types are potential progenitors of Type I core-collapse supernovae (SNe). We used grids of core-collapse models obtained from single helium stars at different metallicities to study the effects of metallicity on the transients and remnants these stars produce. We characterised the surface and core properties of our core-collapse models and investigated their 'explodability' using three criteria. In the cases where explosions are predicted, we estimated the ejecta mass, explosion energy, nickel mass, and neutron star (NS) mass. Otherwise, we predicted the mass of the resulting black hole (BH). We constructed a simplified population model and find that the properties of SNe and compact objects depend strongly on metallicity. The ejecta masses and explosion energies for Type Ic SNe are best reproduced by models with Z = 0.04 that exhibit strong winds during core helium burning. This implies that either their mass-loss rates are underestimated or that Type Ic SN progenitors experience mass loss through other mechanisms before exploding. The distributions of ejecta masses, explosion energies, and nickel mass for Type Ib SNe are not well reproduced by progenitor models with WR mass loss, but are better reproduced if we assume no mass loss in progenitors with luminosities below the minimum WR star luminosity. We find that Type Ic SNe become more common as metallicity increases, and that the vast majority of progenitors of Type Ib SNe must be transparent-wind stripped-envelope stars. We find that several models with pre-collapse CO masses of up to similar to 30M(circle dot) may form similar to 3M(circle dot) BHs in fallback SNe. This may have important consequences for our understanding of SNe, binary BH and NS systems, X-ray binary systems, and gravitational wave transients.