Did the ancient world, (Like ancient greek, maybe when the old Egyptian kingdoms) have to deal with leftover Ice age and stone-age stuff? Like where their tribes of the absolute savage stone age "cavemen" coming upon "civilized" ancient greek cities, or some now extinct giant ice age animals that were still roaming the old forests when Rome was still rising, etc.
Like a guy carving stone for the first pyramids just sees some near-extinct ancient creature that we wouldn't know of because it went extinct in 3000bc, so on.
I cant answer for fauna, but I can about interactions between the types of cultures you list.
Honestly, terms like "Stone Age" and "Iron age" fall apart when you start comparing cultures. They really only have a meaning in parts of Europe, but even then you will see many avoid those terms if possible. An example of this would be cultures in the Americas, such as Middle Mississippians. They built structures, lived in large communities, farmed, and even had forms of art and other abstract imagery. But using the categories you listed, they are "Stone Age" peoples. The whole age system relies on the idea that metalworking is some sort of integral part of being human and it really fits into an unilinear style of human development. Basically it works for the video game Civilization, but not actually for real cultures.
So yes, the Greeks did interact with people who behaved like humans in the "stone age", same way that you or I do. We all are humans from any era you want, humans are still capable of the same feats and mental abilities.