I'm only an undergraduate, and my evidence is mostly literary, so this is going to be a bit of a surface reading, but:

There seem to be common aetiologies for a lot of mythological figures. Pindar's Olympian 1 makes reference to the commonly understood myth of Tantalus, i.e. that he fed his son to the gods, when he puts forth his own interpretation, which is more favourable to his audience:

For me it is impossible to call one of the blessed gods a glutton. I stand back from it. Often the lot of evil-speakers is profitlessness. If indeed the watchers of Olympus ever honored a mortal man, that man was Tantalus.

So in response to the first part of your question, it's entirely plausible that they believed that there had once been a hero called Perseus who killed Medusa. It is asserted in Pausanias that there was a shrine to Perseus on the road from Mycenae to Athens, which is in keeping with the concept of the Greek "hero-cult" - cult worship to unique individuals, located in places associated with that hero (in this case, Mycenae, which Perseus is supposed to have founded)

It's important to expand upon what I mean when I say unique because as I mentioned in another post on this subreddit, "hero" to the Greeks does not mean heroic as we understand it today. Their understanding is more that said heroes did things beyond the realm of mortal man, good or bad - which the Greeks, and many peoples of the time (see: the deification of Alexander the Great by the people he conquered), associated with divinity.

To move on to the second part of your question, this part is tricky because our evidence tends to come from literary figures and playwrights, neither of whom especially represent the 'average view'. However, I think the key lies in the concept of the 'Golden Age', which we find initially in Hesiod (Works and Days, 109-169):

First of all the deathless gods who dwell on Olympus made a golden race of mortal men who lived in the time of Cronos when he was reigning in heaven. And they lived like gods without sorrow of heart, remote and free from toil and grief: miserable age rested not on them; but with legs and arms never failing they made merry with feasting beyond the reach of all evils. When they died, it was as though they were overcome with sleep, and they had all good things; for the fruitful earth unforced bare them fruit abundantly and without stint. They dwelt in ease and peace upon their lands with many good things, rich in flocks and loved by the blessed gods.

This myth recurs throughout so-called 'didactic' poetry through to the Roman era, although it takes different forms (in Aratus' version, for instance, Justice dwells among men, but then leaves when they become bitter). But it is never suggested (not until Virgil's Georgics, and even then that is subject to interpretation) that the current age is a golden one. It is different from how we would understand the 'golden age' today, where it is associated with a specific kind of lifestyle; Greek and Roman literature places it deeply into the past, something unattainable.

To return to the quoted passage, the so-called 'heroic' race, whom Hesiod explicitly describes as the race that went to Troy, are the third descendants of this generation - the men who currently inhabit the earth are the descendants of that. Because Hesiod divides his men into golden, silver, and bronze (Plato draws on this imagery in his Republic), it separates them from the current crop of men.

The point is that the age of heroes and monsters, as represented in both written and dramatic form, is rooted in a mythological past - typically one where the gods dwelt among men - that cannot be obtained. As a result the average Greek or Roman could potentially have been familiar with a great deal of myths, but would perhaps have been less inclined to believe in 'real world' monsters.

With that said, there is a rather infamous passage of Herodotus (3.102) which... well, I'll let it speak for itself: (most historians believe that something got lost in translation)

In this sandy desert are ants, not as big as dogs but bigger than foxes; the Persian king has some of these, which have been caught there. These ants live underground, digging out the sand in the same way as the ants in Greece, to which they are very similar in shape, and the sand which they carry from the holes is full of gold.

In summary I'd say that the age of heroes and monsters turns up rooted in a specific mythological past, and there seem to be common elements that turn up in these even when the overall aetiologies change (e.g: no matter the myth, Medea's children always die, even if she's not the one to kill them), but it should be remembered that both Greeks and Romans had a fluid relationship with the divine and because they were both explorers they often found 'monsters' of their own.