When can ZFC be said to have been "born"? The "History" section of the Wikipedia article on ZFC isn't particularly helpful. The only thing I have understood from it is that ZFC had appeared after 1922. In what book or paper was ZFC first explicitly formulated and proposed?

You might look at the Springer Encyclopedia of Mathematics.  I think it is not so easy to give a definite "birth date" to something such as ZFC, because  many slightly different forms of the axioms appear over a period of several years.
