Just a couple of points to add to the excellent information already given.
As far as I know (and this is "knowledge" gleaned over the years rather than absolute fact) the aborigines have been in Australia for at least 40,000 years. I've never heard the 125,000 year claim but couldn't rule it out.
Originating in Africa the aborigines made their way through Asia and across the land bridge to Oz -- a bridge which now remains only in the form of a string of archipelagos, of which Indonesia is the most prominent.
I really don't see the Australian aborigines being the direct forebears of the American tribes -- they are vastly different from each other -- although it's possible that they shared a common ancestor. If the information posted on this site last week is accurate it's more than likely that this was so.
There were two basic groups of aborigines. One group settled in Tasmania (that's the island across Bass Strait, just south of Australia's east coast) and developed differently to those who settle the mainland where there were many tribes and groups.
The Tasmanian aborigines were wiped out during the first hundred years of European colonisation and Truganini, the last surviving Tasmanian, died in 1876. Our treatment of the aborigines brings no credit to our settlers and, even today, governments (and ordinary people) are struggling to find an equitable balance between the two cultures.
The Torres Strait islanders are another group who also developed differently to both the Australians and the Tasmanians. (Torres Strait lies to Australia's north, between Cape York and Papua-New Guinea.)