I recommend looking at Paul Veyne’s “Did the Greeks Believe their Myths?” Which I think would generally get at your curiosity.

The big thing is that ancient, polytheistic religions worked way differently than modern religions nearly across the board, and the mythologies and cosmologies that supported them were very nebulous.

I think the best answer to your question is “a significant amount” but not in a modern sense. Greek deities/myth were as much “pop culture” as a religion. Pottery and visual art displayed imagery that either told stories or required a certain amount of background knowledge to understand. The Athenians especially (as well as other cities) held drama festivals where different versions of myths and stories were made into spectacles. In other instances, people called aoidoi or rhapsoidoi were itinerant bards who sang the Great works of epic poetry. Think of all you know about your favorite tv shows, movies, video games, and music and then imagine they all took place in the same cinematic universe and then that cinematic universe was also the ethereal and sacred.

To your second point, evidence demonstrates that yes, there was great regional (and class) variety in the knowledge of the people. There are amphorae from certain artists that display a different series of events than from a parallel passage in the Iliad, for example. We know of at least three different versions of the story of Iphigenia. The works of Homer themselves, argue most classicists anymore, were but one version of the story of the Trojan war.* No doubt there was regional differentiation in the quality of knowledge on certain aspects of Greek myth. So yes, it did vary by region and polity not only in how much was known about certain topics, but also what version was known.

The characters, motifs, and gods of myth had to be pretty well-known by the people of the area for their art to make sense, both visual and literary. A great example (though admittedly a little deficient) would be how it’s kinda necessary to know about a show/movie/game to appreciate fan fiction or fanart. While most literature was made by and for the elite and so it should be taken with a grain of salt when making sweeping claims, the volume of visual art and public performance literature shows that this type of knowledge was widespread enough and certainly varied greatly.

*Nagy argues that Homer’s version grew in popularity to become the authoritative version in concert with the rise of Athenian hegemony following the Persian Wars