Is there a common expression for "origin of everything"? What could it be?

If something is a sine qua non, it's absolutely necessary for something else to work or make sense. For example, you could say that walking is a sine qua non for running. This doesn't fit your second sense (origin of origins) very well, though

Genesis is the term generally used:

• the coming into being of something; the origin.

• a beginning or origin of anything

• This chapter will trace the genesis and development of the oldest human civilisation.

also dawn:

• A first appearance; a beginning:
  • the dawn of history.

First Cause:

noun

Philosophy
A supposed ultimate cause of all events, which does not itself have a cause, identified with God.

'Ur', as in the phrase 'Ur example' or 'The Guttenberg Bible is the ur-book'.

[Edit 03/04/2015]

"Forming words with the sense of ‘proto-, primitive, original’."

Source; Wictionary http://en.wiktionary.org/wiki/ur-

"The Ur Example is the oldest known example of any given trope. Ur-' is just a German prefix meaning 'proto-, primitive, or original.'"

Source; TV Tropes http://tvtropes.org/pmwiki/pmwiki.php/Main/UrExample

An "ur-book" would be (one of) the oldest known example(s). In that context, the Guttenberg Bible would not be an 'ur-book' but it would be the ur-example of the printed book.

The ENIGMA machine could be thought of as the ur-computer. It almost certainly wasn't the first functional computer but it is the first example that is generally known. Charles Babbage's analytical engine could also be thought of in this way, as it was conceived at an earlier time, but it was entirely theoretical and no contemporary examples were ever constructed.