Responding to the original 4 questions;
I support the theory of capital punishment even if the practice of it needs improvement.
It does not reflect either old or new covenant because our government is not part of those covenants, it is not a christian theocracy, it is secular and governs a group of mixed religious views. IMO the only way a government can be based on the new covenant, is if Jesus has set it up and runs it personally as king.
If a human gov't doesn't know the person is innocent and kills them with good intentions and good protocol, it's a tragic accident with similarities to manslaughter. But if the gov't knows they are innocent or has not used good intentions and protocol, then it has similarities to murder. But condemning a government isn't something for people to do casually if they have never had to serve in authority over many people or make life/death decisions with dangerous individuals.
"Fair" is subjective and isn't perfect. Innocent life has still been taken and will continue to be taken, no matter how "rare" we think it is. Where we can all agree, innocent people have been executed and that's wrong. Yet, people just have an "oops-sorry-for-killing-you" card when it comes to murdering innocent people that were sentenced to die. "The show must go on..."
I'm against the killing of all innocent life. The unborn and the born alike. I'm not just pro-life when it's convenient for me.
It seems your position is that the life of a few innocents is more important than the death of however many murderers. Given that there are prison escapes and other non-murderous inmates that are vulnerable to violence from these people, you seem to be taking the route that is easiest for your conscience, which is nonintervention. The wrongly-convicted are important, but so are the others who are at risk because of the rightly-convicted, so are the ones who were already murdered by those convicts, and justice itself matters for the sanity of society.
Also, posing this as a pro-life question is disingenuous since you can't be ignorant of how most people use that term. Comparing babies to murder convicts is apples to oranges x1000. The spiritual value of life is inherent, but the right to life is an expression of our agreement on eachothers' value, and when someone takes a life wrongfully they have 'canceled' their agreement.