As for those stoning the woman, Christ did not distinguish between what was a civil matter and what was sin.
He didn't say, according to the account, under what civil or governmental role was she to be stoned, but she was to be stoned according to the laws brought forward by Moses on behalf of God.
It is then Jesus states, he who is without sin cast the first stone. There in lies the message: not the law and whether it was just or legal, but rather who can condemn the woman to death because they are sinless.
He later states that the Pharisees are judging according to the flesh (to condemn her life).
If we jump in at Paul, we are many thousands of years late in discerning a biblical teaching on capital punishment. Gen. 9:6 is where we must begin.
The Bible presents a unified teaching acrooss both Testaments about a govenrment's right and responsibility to execute capitl punishment.
Christ was distinguishing a civil matter from personal retribution. This is in line with the Pharisees misinterpretation of the civil injunction of, "an eye for an eye." Christ also had to correct their changing of that into personal retribution.
Both Testaments teach a difference between personal conduct and responsibilities committed into the hand of civil government.