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You will find very few professional historians trying to tackle huge questions like this. Historians have always specialized their expertise in particular times and places, for good reason. The primary exception to this are the proponents of "big history".
That said, a good place to start is the book Pre-Industrial Societies: Anatomy of the Pre-Modern World by Patricia Crone. Her specialty was in early Islam but she took a broad comparative perspective for this book. She focuses on comparing "complex societies", which includes pre-modern agrarian societies like the ones you mention in the question on the one hand, and modern industrial societies on the other. I will try to briefly address your enumerated questions based on her analysis.
Were most people even aware that they were part of something larger than their immediate social environment (ie, state vs village)?
In order to have anything like a centralized state beyond the village level, you need to have a taxable surplus, typically due to sedentary agriculture. The vast majority of people in human history were outside the preview of such states, living as hunter-gatherers, pastoralists, etc. When such people were aware of agrarian civilizations at all, these people on the margins were the "barbarians". Such people were "dangerous because they knew enough about civilization to both covet and despise it". The exact relationships between civilizations and their barbarians did vary considerably within this framework
Would they care which government to live under given that there was no strong state-related (eg, national) identity?
If we are talking about those people who were actually considered subjects of pre-modern states, we can say that the state "was expected to provide a protective shell behind which the subjects could get on with their own lives, but not regulate their activities or to take over their roles except in so far as enemies of the established order had to be squashed and wrong-doers to be restrained." That said, there were a lot of enemies. Poor peasants were not always happy to pay taxes, and when they dragged their feet, this was not taken lightly. In sum, while pre-modern states did not regulate commoners daily lives in great detail, the routine violence against resistant subjects was considerable.
Would they notice the change of government if territory they lived in was conquered? How fast?
Mostly politics was a very elite affair, concerning "as little as one or two per cent of the population". The main thing that mattered to common people was the frequency and level of taxation. If this did not change, then the specific regime in power generally meant little.
Can anything general be said on this issue, or does it strongly depend on the exact period and geographical location?
Yes, both. If you want more than that, I highly recommend Crone's book. It's easy to find a pirated copy if you are so inclined.