So for example, would an average Greek citizen expect to be able to interact with the gods the way that Odysseus does? Did they actually think that's how the world was set up?
We don't actually know what a "typical" Greek thought about anything. All written sources and most material remains reflect the views and concerns of the rich. We can usually only guess at the extent to which these resembled/emulated/pandered to the views of the common people.
That said, the answer to your question is probably no. The Iliad and Odyssey were set in a remote Heroic Age. They are full of references to the supernatural strength, size and beauty of the heroes - how they could "hurl stones that no three men of today would be able to lift", and so on. The gods took a personal interest in the lives of these heroes because in quite a few cases they had divine ancestry going back just a few generations.
In Classical times, communication with the gods took place through intermediaries like priests, diviners and oracles. To get the attention of a god, you had to make the right sacrifices (which is already true in Homer), and it took specialist knowledge to understand the message you got in return.