Separation of church and state?

It isn't literally included in the Constitution, but it is a short-hand phrase meant to sum up all at once four of the explicit principles of the Constitution: 1) In Article IV these words "The United States shall guarantee to every State in this Union a Republican Form of Government," which means not only that dictatorships and monarchies are prohibited, but also too any kind of theocracy. 2) In Article VI these words: "[N]o religious Test shall ever be required as a Qualification to any Office or public Trust under the United States," which guarantees that anyone of any religion or who is irreligious is allowed to hold office. 3) The First Amendment: "Congress shall make no law respecting an establishment of religion," which means no official religion shall ever be adopted, and that none of the three branches of government may pass on its governmental powers to any church or group of churches. 4) Continuing with the First Amendment: "Congress shall make no law ... abridging the free exercise [of religion]," which means that every single individual in the land has an absolute right to belong to any religion of their choice or belong to no religion at all.

The phrase "separation of church and state" has been attributed to Thomas Jefferson, and the idea also came from Roger Williams long before the adoption of the Constitution.

Because they don't want to be moral; that phrase was actually only in a letter one Founding Father wrote to another; it is in neither the Declaration of Independence, nor the Constitution, and as such has no official standing, as you say.

Because they're trying to get their opinions heard one way or another even if they have to invent lies to do it.

Because the separation of church and state (religion holds no power in politics) is one of the fundamental principles of America, and states explicitly in the constitution.