There may indeed be stories that come to use from beyond the 10,000 year mark, but what is proposed in Australia is, in itself a reach and helps to demonstrate the problem with determining the age of stories from that long ago. Because people move, narratives diffuse geographically, and oral tradition by its nature changes over time, recognizing a narrative as speaking from an ancient time is extremely difficult. The Australian example was possible because the indigenous people of Australia exhibit an amazing amount of cultural continuity and this story in particular can be taken to describe coastal islands that disappeared with the end of the last glaciation. In other words, there were specific motifs that could be dated to a specific time, embedded in stories told by people who had been in the neighborhood for thousands of years. That is extremely rare.

Many of the specific things you ask about either did not happened (there is no evidence of "wars between humans and Neanderthals" (who were also human, by the way), and changes in skin tones were likely gradual enough that people were not likely to have found it to be remarkable. The same was true of most expressions of climate change - our current situation is relatively anomalous.

In general oral histories become extremely unreliable after several thousand years - if not before. The circumstance varies with each place and each culture. If people remain in the same location and if they place a value on repeating stories as heard, then there is a chance for people to tell a story with some integrity based on past events. Because stories do, in fact, mutate, reaching too far back is simply too much to expect in most circumstance. This problem is enhanced by the fact that some cultures welcome changes in narratives and by the fact that people who move from one place to the next tend to forget the original context/meaning of the historical narratives, which cause the narratives to drift all the more.

With all that set aside, let's consider the possibility of stories that may date to 10,000 before present or before. These are not "oral histories," but rather they are stories that may have been told to be believed (or they may represent fictions told to entertain); either way, there is some evidence of integrity in the story telling reaching back millennia. The work of Julien d'Huy is particularly intriguing if not also controversial. He has applied what he calls "phylogenetic reconstruction" using an approach adapted from the study of genetics and mutations to describe stories that have genealogies, which he maintains can be thousands of years old.

One of his best - and oldest - examples is the story of the Cosmic Hunt which d'Huy maintains is tens of thousands of years old based on its wide distribution in Eurasia as well as in the Americas. But there are other examples of stories that he maintains are several thousand years old, something he maintains can be documented with his method. d'Huy's work is focuses on narratives that we would NOT call oral histories, and so it is in some ways easier to track the mutations that occurred and trace them back to a postulated first telling.

As indicated, d'Huy is not without his detractors, but his method is intriguing, and I see no reason to discard what he has accomplished without good evidence that he is wrong (and I have not seen that).

This is not as satisfying to some as the Australian example, but as indicated, the circumstance that make that identification possible is so unusual that it is rarely replicated. There are some stories about the volcanic explosion that produced Crater Lake (7,700 B.P.), but that too is stretching things to the point where it is extremely rare and unusual.