Short answers, pulled from the story.
Early human migrations are believed to have begun approximately 2 million years ago with the expansions of Homo erectus out of Africa. Homo erectus migrated via the Levantine corridor and Horn of Africa into Eurasia around 1.8 million years ago.
Homo sapiens are believed to have emerged in Africa about 300,000 years ago, based in part on thermoluminescence dating of remains from Jebel Irhoud, Morocco, published in 2017. The Florisbad Skull from South Africa, dated to about 259,000 years ago, has also been classified as early Homo sapiens.
Modern humans colonized Australia by around 65,000 to 50,000 years ago by following the Asian coast through Southeast Asia and Oceania. Crossing the gaps of the Weber Line, up to 90 km wide, between the Sunda and Sahul land masses required seafaring skills.
North Eurasian populations migrated to the Americas about 20,000 years ago, after the Last Glacial Maximum, crossing the Beringia land bridge between eastern Siberia and present-day Alaska. Conventional estimates place humans in North America between 15,000 and 20,000 years ago.
Yes, migrating modern humans interbred with Neanderthals and Denisovans, so contemporary populations descend in small part, below 10% contribution, from regional archaic humans. Denisovan DNA makes up 0.2% of mainland Asian and Native American DNA, and Denisovan ancestry is shared by Melanesians and Aboriginal Australians.
Early waves of Homo sapiens reached the Levant by around 185,000 years ago and possibly China by 125,000 years ago, but these migrations receded by about 80,000 years ago without leaving traces in modern genomes. Proposed explanations include replacement by local Neanderthals and the Toba eruption 74,000 years ago.
The Pacific islands were settled by Austronesian peoples originating from Taiwan, who built the first sea-going ships using catamarans, outrigger boats, and crab claw sails. Their Lapita-culture descendants reached Samoa and Tonga by around 900 to 800 BCE, Hawaii by 900 CE, Rapa Nui by 1000 CE, and New Zealand by 1200 CE.