to hang or not too hang

As a Protestant Christian in the US, I am aware that many denominations here are pro-capital punishment. They can quote scriptures to "back up" what they say. I personally have a hard time with it. I remember the story of the thief on the cross and I am reminded that anyone can be forgiven by God and brought to salvation , even in the eleventh hour. I worry about executing people who have never been given the opportunity to be introduced to Christ. This obviously is not the state's responsibility. They are concerned with the safety of the general public and are not in the soul saving business. I also struggle with this situation in light of the commandment "Thou shalt not kill". I know that it is legal here in only certain states (each state has authority to decide this issue). I also know that most of those states (I'm not fully informed here) do this deed by "lethal injection" which is supposed to be the most painfree method. I also struggle with this because euthanasia is something that I am opposed to morally. If capital punishment is allowed, wouldn't euthanasia be a next logical step? Are we not that much closer to it being accepted too? I am not sure where the Catholic church stands on this topic globally let alone in the US. Maybe you could enlighten me.