(I can only speak with any sort of authority on antiquarianism in Britain so that's what I'll focus on.)

As far back as cavemen, almost certainly not, but there was a growing awareness of the fact that people in the past weren't like they were in the present. The understanding of the progression of man is something that can be largely tracked by the development of antiquarianism and the first case in Britain was John Rous (1415-76) (see also William Worcestre (1411-91)). He recorded the difference between historic and contemporary armour styles that he found in Warwick Castle and extrapolated this with varying degrees of success. While his work didn't have a great immediate influence, it went on to impact John Leland (1503-1552) and then William Camden (1551-1623) who would both play major roles in the development of antiquarianism in Britain.

Camden in particular began to speculate on the nature of primitive Britons following the discovery of indigenous people in the New World, speculating that primitive Britons would have looked much like them and used similar tools. Worked flints were also discovered and, as people realised that the flints being made in America were the same as often found in Europe, the connection was slowly made that ancient Europeans were also once at the stage (there were still people who believed them to be fairy arrows and lightning bolts into the 18th century though). However, given that this is the late 16th/early 17th century, it is a stretch to call it still medieval and it should be remembered that this was very much a small elite - even if Camden's work did bring antiquarian findings to a broader audience.

It should be remembered that during the medieval era Geoffrey of Monmouth's Historia Regum Brittaniae (famous for, amongst other things, being behind the popularisation of the King Arthur myth) was largely seen as fact. While there were consistent attacks on the work, it was largely accepted until the mid-16th century when it began to lose favour (Leland was one of the last great fierce defenders of it). We can see repeatedly the nature in which people thought that Arthur's era and the contemporary medieval era were essentially the same, most famously when Henry VIII and Francis I sought to emulate it with the Field of the Cloth of Gold as well as the desire to create castles that reflected Arthurian legend. To me this seems indicative of a people trying to reclaim a golden age of the past, rather than of a people understanding the comparative poverty and primitiveness of their predecessors.

Sources: British Antiquity (1950) Kendrick; Ancient Britons and the Antiquarian Imagination (1989) Piggott. If you want any more, I can happily provide more works that may be of interest.

I don't have my sources on hand at the moment, but the history of archaeology really starts during the Renaissance, with the renewed interest in ancient Greco-Roman culture and architecture. This study was known as "antiquarianism". Archaeology would take many more years to mature and the excavations would have to get more and more sophisticated until it became obvious to archaeologists that they had discovered evidence of what men were like before the Romans in Europe. What you know as a linear history of man is the result of painstaking analysis of what has forever been a "random" assortment of stuff buried in or sticking out of the ground.

In brief: the answer is no. This is because there was no evidence to know what they were like or to suggest that they existed and the doctrine at the time would have been to refer to the bible for the world's history, which certainly has no place for cavemen after the Garden.

I will also add that what many people know as "cavemen" are probably the Neanderthals who often lived in caves for shelter during Europe's Ice Age. Their existence was definitely unknown to people until well into the scientific revolution and stone age homo sapiens actually lived somewhat differently, and arrived to Europe well after the former species (the details of this are probably off-topic).

The first Neanderthal remains were found in a cave in the Neander valley in the mid 19th century. The identification was controversial. Some claimed the bones were those of a Russian soldier who died in the cave during the Napoleonic wars. Others, that it was an old mentally defective man with arthritis.

The "cave man" idea was popular in the late 19th century following the acceptance of Darwin's theory of evolution. Then there was the search for the "missing link" to prove Darwin's theory.

So the idea of cave men was from the late 19th century. There were stories about primitive men in traveller's tall tales and "noble savages" but the cave man was something else.

So far, lots of interesting answers regarding "Western" civilisation. I'm also interested to know about other cultures, particularly the Chinese (as I've read about a great deal of parallel development of philosophy, mathematics etc)

Follow up question, do we know how they treated artifacts and cave drawings from that time. It obviously would differ from place to place but in general.

This answer is going to focus on the early and high Middle Ages, things did change somewhat with the transition into the late Middle Ages and Early Modern period.

The answer is almost certainly not. You have to understand that the classical world (the area around the Mediterranean) there was a constant technological push and pull, with new ideas diffusing or conquering people bringing them, keeping relative parity of technological development over extremely long time spans.

In fact, most of medieval understanding of the way things change as time progresses is not advancement, but decline. This is informed by two major things: Roman ruins and Christianity.

Medieval people lived sometimes literally in the shadows of Roman ruins. Educated people recognized that these were the works of men, but they were still awe inspiring in their size and complexity. The famous phrase from Beowulf "eald enta geweorc" translates roughly to "the old works of giants" - which captures how these ruins were seen. It's possible common people literally thought of them as having been built by giants, but more likely there was an understanding that they were the ruins of a great past civilization. So Medieval people were constantly living around reminders that there was once a much greater civilization living on the same land as they now inhabited that had capacities far beyond their own.

The other thing to remember is that the medieval world view was powerfully influenced by Christianity, which is a story of a great fall with partial redemption. Humanity started in a state of perfection and harmony with nature, fell, then suffered decreased lifespans and magnificence generation after generation. Christ redeemed humanity, but humans must still live their entire lives suffering from their sinful nature and struggling to redeem themselves.

On the second question, Herodotus mentions multiple peoples of various technological states in the Histories. It seems to be one of his main themes and he actually mentions troglodyte peoples in northern Africa.

ORIGINS AND SAVAGE PERIOD OF MANKIND

But mortal man

Was then far hardier in the old champaign,

As well he should be, since a hardier earth

Had him begotten; builded too was he

Of bigger and more solid bones within,

And knit with stalwart sinews through the flesh,

Nor easily seized by either heat or cold,

Or alien food or any ail or irk.

And whilst so many lustrums of the sun

Rolled on across the sky, men led a life

After the roving habit of wild beasts

As yet they knew not to enkindle fire

Against the cold, nor hairy pelts to use

And clothe their bodies with the spoils of beasts;

But huddled in groves, and mountain-caves, and woods

This is two selected sections of the poem "De Rerum Natura" or "On the Nature of Things" by the Roman poet and philosopher Lucretius. Lucretius' work dates to the 1st century BCE, so well before the medieval era. Lucretius goes to describe the invention of agriculture, metalworking and similar things. The other Books of the poem deal with other things, the nature of atoms, the infinite nature of the universe, ect., but this sections deals with humans.

There were other ancient authors who expressed thoughts, but this was the most direct example i could come up with. My point is that ancient romans and greeks knew about cavemen and some of the early developments of civilizations.

As for medieval people, they had the ancient texts, so we must assume that at least some of them knew. There were copies of "De Rerum Natura" in a number of medieval libraries, but only few people would have had access to these. However by the early 15th century people started producing quiet a few copies and thus it became more widely available.

That is at least my perspective as a classical era specialist.

note: the sections of "De Rerum Natura" included in this post stems from the William Ellery Leonard translation, which is available for free through the Perseus Collection here. The lines quoted are 5.925-35 and 5.963-6.