  We perform hydrodynamic supernova simulations in spherical symmetry for over
100 single stars of solar metallicity to explore the progenitor-explosion and
progenitor-remnant connections established by the neutrino-driven mechanism. We
use an approximative treatment of neutrino transport and replace the
high-density interior of the neutron star (NS) by an inner boundary condition
based on an analytic proto-NS core-cooling model, whose free parameters are
chosen such that explosion energy, nickel production, and energy release by the
compact remnant of progenitors around 20 solar masses are compatible with
Supernova 1987A. Thus we are able to simulate the accretion phase, initiation
of the explosion, subsequent neutrino-driven wind phase for 15-20 s, and the
further evolution of the blast wave for hours to days until fallback is
completed. Our results challenge long-standing paradigms. We find that remnant
mass, launch time, and properties of the explosion depend strongly on the
stellar structure and exhibit large variability even in narrow intervals of the
progenitors' zero-age-main-sequence mass. While all progenitors with masses
below about 15 solar masses yield NSs, black hole (BH) as well as NS formation
is possible for more massive stars, where partial loss of the hydrogen envelope
leads to weak reverse shocks and weak fallback. Our NS baryonic masses of
~1.2-2.0 solar masses and BH masses >6 solar masses are compatible with a
possible lack of low-mass BHs in the empirical distribution. Neutrino heating
accounts for SN energies between some 10^{50} erg and about 2*10^{51} erg, but
can hardly explain more energetic explosions and nickel masses higher than
0.1-0.2 solar masses. These seem to require an alternative SN mechanism.
