In the scripture you referenced, Jesus was not repeat not making a statement about capital punishment. As He knew the Pharisees had no interest in adultery other than the chance to force Him into making a stand about Mosaic law or affirming their right to use capital punishment while under Roman rule where the Empire reserved that capability to themselves.
He not only thwarted them, but pointed out their hypocrisy in only presenting the woman ‘caught in adultery’. If they had caught her at it, where was the man? Mosaic law would have them both stoned. By saying ‘those without sin cast the first stone’, He in effect challenged them to put up or shut up regarding the use of capital punishment without the Roman empire’s sanction.
Finally, since Jesus was all about repentance and turning from sin, He didn’t condemn the woman after the Pharisees had balked but instead admonished her to ‘go and sin no more’.
Paul’s letter to the Romans chap 13 affirms rightful authority to administer capital punishment because all authority on earth is derived from God.
The current stance of the Catholic church is anti-capital punishment is it not? The question is at what point in history did it change? They encouraged it use against its enemies in the past.
From the Catechism:
#2267 (LINK) Assuming that the guilty party's identity and responsibility have been fully determined, the traditional teaching of the Church does not exclude recourse to the death penalty, if this is the only possible way of effectively defending human lives against the unjust aggressor. If, however, non-lethal means are sufficient to defend and protect people's safety from the aggressor, authority will limit itself to such means, as these are more in keeping with the concrete conditions of the common good and are more in conformity to the dignity of the human person.
Today, in fact, as a consequence of the possibilities which the state has for effectively preventing crime, by rendering one who has committed an offense incapable of doing harm - without definitely taking away from him the possibility of redeeming himself - the cases in which the execution of the offender is an absolute necessity "are very rare, if not practically non-existent."
The last paragraph ("Today, in fact...") is a prudential judgment, not a universal doctrine. That is, an attempted assessment of particular times and places where the severity of capital punishment could be foregone by the substitution of, for instance, life imprisonment without possibility of parole.
Many have noted, with reason, that even a murderer with a life sentence can still aggress against others, e.g. by assault upon a guard or another prison staffer or another prisoner, and that these cases there would be no other recourse except the death penalty.