What is the Catholic view of separation of church and state in terms of war and application of the death penalty? [closed]

Note: This question was an honest search for a Catholic view not an accusation or attack. If I had written an attack it would be a 1000 times more severe and to a specific point. The defensive response in some of the comments leads me to agree that I should have specified the question more narrowly. However I had to ask my real question to get the answer I was looking for. I will now have to also vote to close the question as it seems a little to complicated for the stack exchange guidelines. Thanks to the accepted answer, and the considerable effort it takes to provide one. I know how it feels to he on the other end, and sometimes this community gives better quality answers than can be obtained elsewhere and the person making the question can be very ungrateful if it conflicts with their own faith.

I have never encountered a Protestant leader who believed at any point in history that the church should take up arms, acting as a sort of temporary civil power, and engage in military conflict or apply the death penalty to heretics. I'm sure there might be some odd exception to the rule but, in general, Protestant churches hold separation of church and state. Where there may have been any direct official church involvement in any such action in old Lutheran or Anglican or whatever churches, Protestants in general would unanimously denounce those actions, as the church is separated from state functions. Some Protestants think it's acceptable as a 'citizen' to support the government on civil grounds to war, or even to apply the death penalty, but not as a church. At least this is the most common Protestant view.

However, I was surprised when I looked up the Catholic Encyclopedia on the Inquisition. It more or less describes it as something that was necessary for the church to do at the time and does not seem to be opposed to that period when Popes led military conflicts and applied death penalties to perceived heretical Christians, even as they took the clear lead.

So it really leaves me wondering: Does the Catholic Church still think such things are OK? Is it OK that at times in history, or potentially the future, Catholics will bear the sword and become a civil power rather then a spiritual one only?

I really mean is it considered OK at times to 'take the clear lead' in commanding others to kill people, as they did when killing people through the crusades and bulls against perceived heretics. I do not mean merely Catholic citizens following their kings or government, but the 'command' of those kings or governments 'to kill.' Does the Catholic church separate church and state or assume it is a state with power to declare war and kill if it seems God's will?

I always thought the whole world no longer approved of that, but now am thinking maybe because Catholics can't admit Papal errors they are sort of stuck approving all those ugly things done in the past by their Popes, but am not sure if my instincts are wrong. Maybe Catholics do oppose direct church involvement in killing others. I really am curious to find a reliable answer on this. I would like to know if Protestants and Catholics generally have a different view.