Criteria:
Exodus 18:21
Moreover thou shalt provide out of all the people able men, such as fear God, men of truth, hating covetousness; and place such over them, to be rulers of thousands, and rulers of hundreds, rulers of fifties, and rulers of tens:
Notwithstanding kingly zeal not for one's own power, prerogative and discretion but for God's law:
Psalm 119:139
My zeal hath consumed me, because mine enemies have forgotten thy words.
But the man of authority should wield his authority under authority.
Matthew 8:9
For I am a man under authority, having soldiers under me: and I say to this man, Go, and he goeth; and to another, Come, and he cometh; and to my servant, Do this, and he doeth it.
God allows all kind of tyranny to work his holy will but he is jealous of his glory:
Acts 12
21 And upon a set day Herod, arrayed in royal apparel, sat upon his throne, and made an oration unto them.
22 And the people gave a shout, saying, It is the voice of a god, and not of a man.
23 And immediately the angel of the Lord smote him, because he gave not God the glory: and he was eaten of worms, and gave up the ghost.
Because I like Newt, I would not want to give him any more temptation in his life. He has been an unfaithful servant. You would not draft a person eaten up with cancer to champion your team in the NFL.
Luke 19:17
And he said unto him, Well, thou good servant: because thou hast been faithful in a very little, have thou authority over ten cities.
However, to cite one example from your list, if I believe “fearing God” is an absolute requirement for a candidate in a civil election, I have an immediate problem in this election because the confessions of my church say some pretty seriously negative things against the Roman Catholic Church. I can say what I believe to be true, namely, that there are sincere Christians in the Roman Catholic church who are sincerely wrong, but I can't get around the confessional standards. Gingrich would not be allowed to come to communion in my church, I would not be allowed to do so in his parish, and there are important reasons for that. The Council of Trent said some pretty serious things about us Protestants, too.
There is no major party candidate running this year, with the possible exception of Michele Bachmann, who would have been acceptable to John Calvin or most other leaders of the Calvinist wing of the Reformation — and in her case, Knox would have said some severely negative things about the “monstrous regiment of women” so even Bachmann wouldn't have been acceptable to the founder of Scottish Presbyterianism as a civil ruler.
I'm not a full-blown Kuyperian, but I believe Abraham Kuyper was correct in saying that within the sphere of the church, we need to adhere strictly to the confessions, but because the sphere of the state is not (at least in most modern nations) formally covenanted with God, Christians can cooperate with people in the state who we would never let into church office.
Kuyper fleshes out in theology the way most of the Reformers worked in practice. How did the Reformers actually work in city-states like Zurich, Geneva, and Basel, and how did they work in larger nations like the Netherlands, Hungary, France, Scotland, and England? Henry VIII was far from the only Protestant ruler to have a moral life that makes Newt Gingrich look rather good. The Netherlands would have remained Roman Catholic with tens of thousands of dead Protestant bodies continuing to pile up in the streets if it were not for urban burghers who cared much more about money than God, and if it were not for William of Orange whose background seems fairly close to that of Newt Gingrich.
I wish Newt Gingrich had been discipled by confessional Calvinists or had returned to a confessional version of the largely nominal Lutheranism of his childhood. I think, in reading Gingrich's religious background, that he was served very poorly by the “cheap grace” of much of modern American evangelicalism. However, from what I read of Gingrich's faith, it seems that what he liked about Roman Catholicism is the same thing that has drawn many other neoconservatives into the Catholic orbit — a full-fledged doctrine of church and society which, while wrong, provides an answer to questions that many evangelicals are asking and to which they're not getting answers.
Can I vote for someone like that? Not as a first choice, but at this point my only alternative is Romney or going third-party, and those aren't choices I want to make since they're conceding the 2012 election to Obama.
Of course, we can and probably will agree to disagree. I have a group of Ron Paul supporters in my own church who I have argued against regularly, and that doesn't mean I think they're bad people; I just think they're wrong.