Constantine is the #1 reason I could never ever call myself a Christian, and more importantly he has zero to do with our Governments separation of church and state. So other than saying he is a freak of nature, I will move on to your second part here.
I am some what surprised that you knew about the Danbury Baptist church and the letter from Thomas Jefferson. Most Libs I have asked do not know, even history teachers. They all assume it’s in the constitution some place. I am not sure though if you realize the letter Jefferson wrote the church was a reply to the letter the church wrote him.
The church wrote him expressing concern with this new constitution thing he and the government had implemented. They expressed concern that they may one day be forced to become one faith like they had in England. (The church of England) they did not want that at all. They wanted freedom of religion that they had not enjoyed in England.
In Jefferson’s reply he said “Believing with you that religion is a matter which lies solely between man & his god, that he owes account to none other for his faith or his worship, that the legitimate powers of government reach actions only, and not opinions, I contemplate with sovereign reverence that act of the whole American people which declared that their legislature should make no law respecting an establishment of religion, or prohibiting the free exercise thereof, thus building a wall of separation between church and state.”
Jefferson never said or implied that Church can have nothing to do with government, but that only government can not get involved in church. Actually it was the oposite of what we see today due to a court case in the 1940's.
It is no exaggeration to say that on Sundays in Washington during the administrations of Thomas Jefferson (1801-1809) and of James Madison (1809-1817) the state became the church. Within a year of his inauguration, Jefferson began attending church services in the House of Representatives. Madison followed Jefferson's example, although unlike Jefferson, who rode on horseback to church in the Capitol, Madison came in a coach and four. Worship services in the House--a practice that continued until after the Civil War--were acceptable to Jefferson because they were nondiscriminatory and voluntary. Preachers of every Protestant denomination appeared. (Catholic priests began officiating in 1826.) As early as January 1806 a female evangelist, Dorothy Ripley, delivered a camp meeting-style exhortation in the House to Jefferson, Vice President Aaron Burr, and a "crowded audience." Throughout his administration Jefferson permitted church services in executive branch buildings. The Gospel was also preached in the Supreme Court chambers.
http://en.wikipedia.org/wiki/History_of_religion_in_the_United_States
Again here I am not intereseted in what other countries do. I just mean in America, but thank you for the added information.