It might be best to address the last point first—namely whether there was some idea of the gods which was held in Hesiod that was common to all Greeks.
Greek culture covers the period from Homer to Emperor Justinian's closure of the Platonic Academy in 529 AD, a period of over two thousand years. The Greeks over these two millennia included illiterate peasants in Boeotia and sophisticated philosophers in Athens and Ionia, poor slaves and wealthy aristocrats, Spartan helots and colonists in Sicily, men and women, and democrats and oligarchs. There are few views likely to be held by every single one of this diverse group of people.
Hesiod is somewhere between these the extremes of local rural traditions and urban sophisticated philosophical thought in presenting the gods of traditional myth and ritual in a somewhat coherent and rationalized form, with strong Mesopotamian influences. His work is often described as marking a transition from myth to theology or philosophy in its discussion of the gods.
In Hesiod, as in much epic and art, the gods were portrayed as anthropomorphic. There is some doubt as to the degree to which these stories about anthropomorphic gods were believed literally versus the degree to which the stories were seen as metaphors for forces of nature or accounts of rituals, as Paul Veyne has argued.
Both poems open with invocations to the Muses and thus emphasize the important role of poetry in telling the stories of the gods; the epic poet claims divine inspiration and the authority of the Muses rather than personal authority as a narrator.
Zeus is portrayed as the king of the gods and the father of the Muses in the opening of Works and Days:
Muses of Pieria who give glory through song, come hither, tell of Zeus your father and chant his praise.
A more expansive version of this relationship and its importance is found in Theogony starting with line 36:
let us begin from the Muses who in hymning their father Zeus.