The closest that you'll come is probably Hesiod's Theogonia. Technically, the text is the muses singing hymns of the gods (And again, they chant the race of men and strong giants, and gladden the heart of Zeus within Olympus, -- the Olympian Muses, daughters of Zeus the aegis-holder), but the corpus deals with the origin of the world and the successive generations of the gods.

Like most religions, there were variations between the myths from place to place. The ones set down by Hesiod eventually came to be the most widely-distributed versions. By the time Herodotus came around and wrote his Histories, they were pretty much the common version.

You might find this older thread of interest. There are probably a few others as well if you search cunningly.

The short answer is no. This is because the religious affinities of Greek mythology were to local cults, not to a grand unified corpus of religious doctrine. Each individual local cult might well have its own religious texts, and I mention one in the other thread (the Hymn to Demeter). But efforts to synthesise the diverse religious traditions of the Greek world -- like the Hesiodic Theogony -- were an artificial veneer, with absolutely no link to anyone's religious practice. The myths did eventually become more or less codified, but for antiquarian, not religious, purposes.