  The majority of massive stars are expected to exchange mass or merge with a
companion during their lives. This immediately implies that most supernovae
(SNe) are from such post-mass-exchange objects. Here, we explore how mass
accretion and merging affect the pre-SN structures of stars and their final
fates. We use the stellar evolution code MESA, infer the outcome of
core-collapse using a neutrino-driven SN model, and apply a rapid-accretion
model. Our models cover initial masses from 11 to 70 Msun and the accreted mass
ranges from 10-200% of the initial mass. We find that mass accretion in
particular onto post-main-sequence (post-MS) stars can lead to a long-lived
blue supergiant (BSG) phase. In comparison to genuine single stars, post-MS
accretors have small core-to-total mass ratios, regardless of whether they end
their lives as BSGs or cool supergiants (CSGs), and they can have genuinely
different pre-SN core structures. As in single and binary-stripped stars, we
find black-hole (BH) formation for the same characteristic CO core masses M_CO
of ~7 Msun and >13 Msun. In models with the largest mass accretion, the
BH-formation landscape as a function of M_CO is shifted by about 0.5 Msun to
lower masses. We find a tight relation between our neutron-star (NS) masses and
the central entropy of the pre-SN models, suggesting a universal relation that
is independent of the evolutionary history of stars. Post-MS accretors explode
both as BSGs and CSGs, and we show how to understand their pre-SN locations in
the Hertzsprung--Russell diagram. Some BSGs that avoid the
luminous-blue-variable (LBV) regime are predicted to collapse into BHs of up to
50 Msun while others explode in supernovae and eject up to 40 Msun, greatly
exceeding ejecta masses from single stars. These masses can be even higher at
lower metallicities, and they may fall into the pair-instability-supernova mass
gap. [abridged]
