I've been watching this and thinking about it since I saw the OP.
I can't give all the answer I would like to give, but here's a start. Whole libraries have been written on the relationship between church and state and people have debated that subject for centuries.
Samuel Rutherford wrote Lex Rex (the Law and the King) for the King of England back in the 1600's to explain what the relationship should be between the church and state. The king later had the book publicly burned.
"The book defends the rule of law and the lawfulness of defensive wars (including pre-emptive wars) and advocates limited government and constitutionalism in politics and the "Two Kingdoms" theory of Church-State relations (which advocated distinct realms of church and state but opposed religious toleration)." (Wikipedia, http://en.wikipedia.org/wiki/Lex,_Rex
When Joshua was standing looking over the promised land on the eve of the conquest (Joshua 5) he saw a man with a drawn sword in his hand. As a commander, he asked this warrior "whose side are you on, ours or theirs?" The warrior answered in effect, that he asked the wrong question. The warrior was the commander of the Lord's army. "Are you on my side?"
What denomination this country would be is perhaps the wrong question. There were no denominations in the New Testament church, although Paul chastises the Corinthians because some claimed to be "of Paul", others of Apollos, and others of Christ only. What there was however, was a group of people dedicated to spreading the good news of the Gospel, to carrying out the Great Commission.
Jesus Christ is described in the Bible as being prophet, priest, and king. But He himself said to Pilate, "My kingdom is not of this world." His kingdom is one designed to bring about what we pray for when we ask for "Thy Kingdom come, Thy Will be done on earth as it is in Heaven."
Christ's Kingdom has been described as a Mediatorial dominion, designed by God to be the channel or conduit by which He works in the world, through nature, the individual, family, church, and the state. God made Christ head over all things for the church...Ephesians 1:22.
Long story short, in an ideal world there would not be any particular religion or denomination. There would be a nation trying its best to find and put into practice God's standards for morality, honesty, justice, business accumen, care for each other, right use of scientific discoveries and knowledge, and any other issues that come up. The Bible talks a lot about each of these things. For starters, read through Proverbs. God invented man; He wrote an user's manual, called the Bible.
God is not interested in suppressing society or a country. He wants people to live a full, prosperous, fruitful, and enjoyable life.
There are a many historical examples of what happens when the state tries to become the church, and when the church tries to become the state. Communism has become the religion for many in this world; Nazism substituted government for God. So did the Roman emperors.
We can see the turmoil and repression in many middle east countries today where a form of religion is trying to rule the state. But that is religious people trying to run a country, not a country trying to find what God wants for it.
Apologies for the long post, but, well, 45Auto, you asked a big question. I'm only suggesting some ideas and can't begin to do justice to your question. Thanks for asking. My brain is beginning to hurt from thinking about this.