They are already separated. One is merely a declaration of intent to be free of English control. The other is the Law by which America is governed.
No. It was established to set up a government. It lacked any guarantees of liberties. That's why they added a Bill of Rights later. And the decision to avoid any mention of God whatever was mainly the doing of Jefferson, Madison, and Washington, who saw that as a European corruption they did not want in the new nation.
It didn't equate to any religious idea at all. They merely wanted a "wall of separation between church and state", as Jefferson and Madison explained.
Freedom from religious compulsion of any kind, and freedom of religion from government interference. Which is not to say that they were always as good as their stated principles. The Alien and Sedition acts, for example would have been loved by Donald Trump as they criminalized criticism of the government. But over time, we have come closer and closer to the ideals of liberty and individual rights and responsibilities.