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?
 A: Since my comments on the comment section were getting a little big, I decided to expand the issue on first-order logic into an answer of its own. But first, let me preface this with a caveat, due to a paper by Gregory Moore ("Historians and Philosophers of Logic: Are They Compatible?"): it seems that establishing priority claims in cases such as this is extremely difficult, and not all too relevant. Perhaps a more interesting task is to be found in the establishment of the relation between the ideas of the many authors of the period, and how they jointly contributed to what we now know as ZFC.
Now, based on a reading of Moore's and Ferreirós's work on this, the picture seems to be the following: Zermelo initially proposed his axiom system, in a second-order guise. Later, Fraenkel proposed, rather half-heartily, the adoption of Replacement as an axiom; however, it's first real advocate with an audience was von Neumann, who also proposed Foundation. Independently, Mirimanoff also studied the well-founded portion of Zermelo's system, as well as his own version of Replacement, yet his work went largely unnoticed. At the same time, Skolem suggested the use of first-order logic as a way of making sense of Zermelo's "definite properties", and also seemed to employ a form of primitive recursive arithmetic as his meta-theory.
The issue about who proposed first-order logic as the right logic to axiomatize set theory is a bit muddled, in large part due to the point, discussed above, that there was no clear separation between first-order logic and higher-order logic at the time (the first two decades of the 20th century). In particular, it seems that the first to clearly isolate the first-order part of logic as being worthy of independent study were Weyl (largely due to his concern with predicative systems), some time during the 1910's, and Hilbert, in his lectures on logic in 1917. However, even then the importance of first-order systems was not clearly perceived, so when Skolem proposed to formalize set theory as a first-order theory, his proposal was met with skepticism. 
Interestingly, it seems that von Neumann's papers in the 1920's were a bit ambiguous on the issue. Ferreirós (Labyrinths of Thought, p. 373) writes: "His systems of the 1920's (...) seem to be intended as first-order, and certainly are formalizable within that frame. If that was his intention, von Neumann was the first mathematician to accept Skolem's (and Weyl's) views". If true, then von Neumann's 1928 paper ("Über die Definition durch transfinite Induktion und verwandte Fragen der allgemeinen Megenlehre", which is reprinted in the first volume of his Collected Works), which contains what is, according to Ebbinghaus, the first printed reference to the "Zermelo-Frankel" axiom system, would be a good candidate for the first first-order presentation of ZFC. In any case, it seems that, by the 1920's, this particular issue was not yet decided. It was only after Gödel proved his completeness and incompleteness theorems (so in the beginning of the 1930's) that first-order logic began to have a prominent place among other logic systems, which created a more hospitable environment for Skolem's proposal.
Indeed, in the first of a series of articles, Paul Bernays, introducing what would become known as the NBG theory, already emphasizes that adopting a first-order presentation "considerably simplifies" the resulting system. So by 1937 (the date of publication of Bernays's "A System of Axiomatic Set Theory - Part I"), Skolem's position was much more widespread. If I'd have to bet, I'd say that the first list to contain explicitly all the ZFC axioms in a first-order setting probably appeared during this time (perhaps in some textbook). Another possibility is Skolem's 1929 article, "Über einige Grundlagenfragen der Mathematik", which seems to be more detailed than earlier expositions, but which I have't had the opportunity to check out yet (my university's library has it, but I'll only go there on Tuesday).
Anyway, that's what I've been able to gather as of now. If I find anything else, I'll update this answer.
A: The following should be a comment, because it doesn't attempt to say who first wrote down the ZFC axioms and called them ZFC, but it's way too long.  I'd better begin by quoting, from the historical appendix in Peter Freyd's book "Abelian Categories": "The origin of concepts, even for a scholar, is very difficult to trace.  For a nonscholar such as me, it is easier.  But less accurate."  With that preamble, here's how the history looks to me.
Zermelo's 1908 axiomatization differed from ZFC in three ways. (1) It used the vague notion of "definite property" in the separation axiom.  (2) It lacked the axiom schema of replacement. (3) It lacked the axiom of foundation (also called regularity).  
Concerning (1), ZFC uses first-order definability as a substitute for (Zermelo would say "as an approximation to") definiteness.  As a result, separation is not a single axiom but an axiom schema.  As far as I know, this change was first proposed by Skolem.  
Concerning (2), I believe Fraenkel proposed replacement as an additional axiom (schema), but Skolem may have also proposed it independently.  It is the reason for the "F" in "ZFC".
Finally, concerning (3), the situation is not very clear to me.  I understand that the concept of well-founded set was studied by Mirimanoff in (I think) 1917.  I don't know, however, whether he proposed as an axiom that all sets should be well-founded.  Von Neumann did propose that, but I don't know who else (besides Mirimanoff) might have done so earlier.
By the way, Zermelo seems to have accepted the need for replacement and foundation, but he didn't like Skolem's "first-order" idea at all.  (My impression is that he didn't like Skolem at all and viewed him as a trouble-maker messing up his nice axiom system.)  In "Grenzzahlen und Mengenbereiche" (1930), Zermelo bases his axiomatization on what we would now call an infinitary logic.  He also uses a version of the axioms that allows for atoms (also called urelements), and he describes what are sometimes called the natural models of set theory: The cumulative hierarchy, of any inaccessible height, over any set of atoms.  The height and the cardinality of the set of atoms are the two parameters that determine the model up to isomorphism.  His use of infinitary logic allows him (if I remember correctly) to exclude any other models.
A: Here are some relevant quotes from Fraenkel, Bar-Hillel, Levy's  "Foundations of set theory":
"Zermelo's vague notion of a definite statement did not live up to the standard of rigor customary in mathematics ... In 1921/22, independently and almost simultaneously, two different methods were offered [by Fraenkel and Skolem] for replacing in the axiom of subsets the vague notion of a definite statement by a well-defined, and therefore much more restricted, notion of a statement ... The second method, proposed by Skolem and, by now, universally accepted because of its universality and generality ... It [The axiom schema of replacement] was suggested first by Fraenkel and independently by Skolem."
An english translation of Skolem's "Some remarks on axiomatized set theory" appears in Heijenoort's "From Frege to Godel - A source book in mathematical logic". The commentary on Skolem's paper says (and I agree): 
"These indications do not exhaust the content of a rich and clearly written paper, which when it was published did not receive the attention it deserved, although it heralded important future developments".
A: Reading in Kanamori's introduction to the Handbook of Set Theory, the text seems to support Robert Israel's answer.
On p.10 it says that von Neumann wrote in his Ph.D. dissertation an axiomatization of set theory, however it is not clear to me (from this text, and from other texts that I have seen this mentioned) whether or not his was the basis for the $\sf NBG$ set theory, rather than $\sf ZFC$ itself.
On pp.11-12 it says that in 1930 Zermelo published an axiomatization which was $\sf ZFC$ without Infinity and Choice (so perhaps $\sf ZF$ would be better here). So if we include those back into the axioms, it works out fine. There is no citation, but I suppose that sort of limits the possible papers.

The paper referred to by Kanamori is the following,

Zermelo, E., Über Grenzzahlen und Mengenbereiche, Fundamenta Mathematicae 16 (1930), no. 1, pp. 29--47.

And indeed on the second page of the paper it says "ZF". Kanamori indicates, as I wrote, that the axioms were missing both choice and infinity. However in his paper (which shares similarities to the introduction of the Handbook I was using before)

Akihiro Kanamori, The mathematical development of set theory from Cantor to Cohen, Bull. Symbolic Logic 2 (1996), no. 1, 1--71.

Kanamori points out that Zermelo assumed the axiom of choice as part of the logic. It is not fully clear to me whether this axiomatization was a first-order or second-order one, perhaps someone that can read German better than me can help with this part.
Once moving to first-order logic, the addition of choice and infinity into the list becomes inevitable, and we have the modern axioms of $\sf ZFC$.
A: 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.
