For example, did people living in Ancient Rome think of its culture, fashion, and traditions as relatively static or did they realize how these things have changed and will continue to change? I know that the republic itself was a symbol opposing the monarchy of the past but to my knowledge, those kings were semi-legendary, so this seems somewhat similar to a creation myth.
On a broader timeline, during the Italian Renaissance, when people were revering the greatness of Ancient Rome, how anachronistic were they in their world views? Did they ascribe their own contemporary cultures to past societies?
One such example of this can be seen in the Ancient Greek Historian Polybius's Histories. In this work, Polybius argues that civilizations, just as living beings, had life cycles, and that these life cycles could be well defined. He states in Book 6, Chapter 4 that "kingship; which degenerating into the evil form allied to it, by which I mean tyranny, both are once more destroyed and aristocracy produced. Again the latter being in the course of nature perverted to oligarchy, and the people passionately avenging the unjust acts of their rulers, democracy comes into existence; which again by its violence and contempt of law becomes sheer mob-rule."
Not only does Polybius illustrate a belief that societies change, he even puts forth a path which these societies follow. Indeed, he uses this idea of a lifecycle of society to explain Rome's supremacy over Carthage, stating in Book 6, Chapter 51, that "For exactly so far as the strength and prosperity of Carthage preceded that of Rome in point of time, by so much was Carthage then past its prime, while Rome was exactly at its zenith, as far as its political constitution was concerned."
For further reading, see Polybius's Histories. Especially Book 6. I personally recommend Tufts' Perseus for a good, easily navigable online translation. The link is: http://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A1999.01.0234%3Abook%3D1%3Achapter%3D1