Q:

Isn't acknowledging the existence of God, as a state, a contradiction of the separation of Church and State?

I'm not an expert in politics at all, but I have been wondering about this for a while.
If Church and State are supposed to be separated, isn't it a bit of a contradiction to have "God" so strongly "embedded" in politics? I'm thinking of:

In God we Trust; official motto, printed on money.
[...] so help me God; president swearing-in.
[...] God bless America; pretty much closing every official announcement.

And so on. 
How is explicitly acknowledging to believe in God, as a state, not being something that intrinsically goes against the principle of separation?

A:

No, it isn't a contradiction under US law.   This has been tested in the Federal Courts, see for example O'Hair v. Blumenthal, and Aronow v. United States. The basic reasoning is summarized in this paragraph from the Anonow case:

It is quite obvious that the national motto and the slogan on coinage and currency "In God We Trust" has nothing whatsoever to do with the establishment of religion. Its use is of a patriotic or ceremonial character and bears no true resemblance to a governmental sponsorship of a religious exercise.

Religion and the government have a lengthy history in the U.S.   Thomas Jefferson's position was that no man should be compelled to support religion, but that all men should be free to profess their opinions in matters of religion.  In context of today's politics and law, the profession of God in those contexts isn't compelling religion, but fits closer the freedom Jefferson described.  In the US today, it is optional.
So, U.S. presidents have never been required to add "So help me God" to their oath, but it became a tradition.  Some U.S. state constitutions required it, and that wasn't viewed as a conflict in 1776, but the law evolved, and in 1961 the Supreme Court ruled against a state constitution that compelled the words "So help me God" in a state oath of office.  The Supreme Court ruled compelling this unconstitutional and unenforceable but did not make optional words illegal.   
Thomas Jefferson avoided the word God in official context, but professed in words of his own ending his first inaugural address:

And may that infinite power, which rules the destinies of the universe, lead our councils to what is best, and give them a favorable issue for your peace and prosperity.

A:

Disclaimer: I'm not that versed in history, so the following might have factual mistakes.
To understand the source of the American separation of church and state, you'll have to take a dive into history.
Between the 16th and 18th century, Europe was in a state of religious turmoil. The ruler decided which religion his subjects would have to follow (Cuius regio, eius religio (thanks @molnarm)), so if you were born under a Catholic ruler and your village was then conquered by a Protestant, you'd have to convert or be persecuted, banished or worse. Just paying lip service to a religion could also have dire consequences. If you then got re-conquered, you'd have to convert again. Due to pretty much constant wars, such a situation wasn't too uncommon.
America became a safe haven for people where they could live without anyone forcing their religion on them, which is why many victims of religious persecution and bloody wars chose to migrate west. The thought of "We don't want anyone to tell us which religion we must have" became ingrained into minds of the American population, but it was still assumed that everyone was a Christian. When the USA achieved independence, the founders decided to formalize it so that a situation like in Europe would never happen on American soil.
The American separation of church and state is not about separating religion from politics, it's about separating the citizens from the religion of the ruler. The USA was still founded as a pretty much Christian nation, but the founders wanted to guarantee that no citizen would be forced into a specific denomination on the whim of a ruler.
In the general case, yes, acknowledging a god as a state means that you discriminate against polytheist churches and atheist belief systems. You connect the state to all monotheistic churches. However, many European and American countries were formed or are based on countries formed during a time when it was assumed that everyone was some sort of Christian and there wasn't enough political pressure for stricter separation and so the connection between many American and European states and Christianity in general still exists.
As a side note, the oath that a president has to swear doesn't have to include the Bible. The president can choose what he swears on. It's just that most are Christians and decide to swear on the Bible.

A:

"Separation of church and state" is not actually required by the US Constitution.
Wikipedia says that the phrase "separation of church and state" was first used by Thomas Jefferson in this letter in 1802, when he was President:

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,
  & 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 & State.

Jefferson's use of the phrase indicates that it is the effect of the First Amendment restrictions on the government, but it is not the literal language.  Justice Hugo Black wrote that the phrase expresses the intent of the literal language of the amendment.
This matters a great deal.  A court case concerning the First Amendment should look at whether a particular issue involves an establishment of religion, or a prohibition of the free exercise of religion.  But it is actually not relevant whether a particular issue involves, or violates, separation of church and state - because there is no such requirement in the Constitution!
Usually the distinction is minor, but it can be important.  One example, that OP mentioned, is the use of "In God We Trust" on US bills and coins.  It's pretty obvious that this is not compatible with the idea of separation of church and state.  But that doesn't actually matter.  It does not establish religion, and therefore it is permissible according to the Supreme Court.
There is considerable legal debate about this point, though, as described in the previously linked paper.  The divide along liberal/conservative lines is basically around whether the First Amendment requires the state to be secular, or simply requires that it not give advantage or disadvantage.

Criticism of the modern Court’s separationist approach has existed since the 1940s, but gained momentum as a result of the resurgence of conservatism during the 1980s and the appointment of constitutional conservatives to the Supreme Court. Today, it is not uncommon for religious, legal, and cultural conservatives to criticize the concept of church-state separation. Critics charge that a separationist perspective imposes a regime of secularism, one that is not neutral toward religious matters but that privatizes and marginalizes religion. Yale law professor Stephen L. Carter argued that the separationism promoted “a culture of disbelief,” while Catholic theologian Richard John Neuhaus claimed that it created a religiously “naked public square.

