Intentions matter, but actions more. The state executes people to keep private citizens from exacting revenge. Due process allows the accused to defend himself, but the very process has become so corrupted that it is in too many cases, justice denied.
Never heard that one either. I don't think that's a known reason in criminal law for capital punishment, but I think it's imaginative and creative. But it fails because the same facilities that keep a dangerous criminal separated from society also keep revenge seekers on the outside from getting to the convict. How these matters are handled within the prison system is a whole other discussion.
That's the exact opposite of what should happen. Once the court has sentenced them, they should turn them over to the appropriate civilians for punishment.