Thank you so much for that brilliant post, Nikolaos! I have actually pondered upon that issue lately - about Socrates's belief or not-belief in Greek gods.

As you stated, Nikolaos, it seems to turn to the "Apology." Some further grounds to believe that in this text Socrates's beliefs are presented as they would have by the person behind Plato's character, are of course that this was a dialogue from the 390's, and presumably Plato's first work to the public. This does not mean it is word-to-word what Socrates said when defending himself (with Plato present) but that it most likely is not very far - later dialogues, for example, "The Republic," do not include any Socrates, I might claim (meaning that Socrates did not envision political systems of that kind).

In many, I'd say most, of the dialogues - as in the example given by Nikolaos - there are references to "gods" or "daimon(ion)s" or such. I think those are normally viewed as "manners of speak" - we know how the Greeks liked to start their clauses by "O Zeu," or "O'nthrope," I mean, with both reference to the gods and humans :-) So the matter stands quite undecided in the rest of the dialogues, except for the myths Plato presents (for example the ending myth of "The Republic"). But mythos, as logos, was one way to present something to the reader (which, by the way, of course makes it harder for us to interpret the text).

(Just on a quick note, I earlier made one tiny post about "genii" and "daimons" here.)

In "Apology," Socrates announces his faith in the effects of daimons, and as such, in daimons themselves (27a-e). Socrates also bases his firm belief in his wisdom - the fact that he knows he does not know - in the words of the Oracle at Delfoi (20b-e).

To me, it seems Socrates was also a good bit agnostic, and especially Plato was such. Famously in the "Meno" and "Phaedo" Plato bases his thinking on the doctrine of anamnesis, and attempt to prove the immortality of the soul. But it seems Plato's theses prove only half of it - they prove it might be so, the soul can be conceived of as immortal. But the opposition still raises in those dialogues, and especially in "Phaedo," one of the most beautiful pieces of literature ever written, Plato's Socrates is all but certain about his firm belief.

Socrates talks about daimon kaina or daimonion in the "Apology" and many other dialogues. It seems more reasonable, to me, to think that he was a sort of agnostic, who did not strictly speaking believe in the Greek pantheon - we might also ask if people normally believed in the gods in those days - but still held a belief in something "supernatural" and extraordinary, something that spoke to him himself.