Just to address one aspect of your question, in North America Europeans did not quickly outnumber native populations. First contact events occurred along the Atlantic and Gulf coasts in the early 1500s, and for more than a century Europeans maintained only tenuous footholds on the continent. To put this into perspective, a century after initial contact more than two million people lived east of the Mississippi River. Less than five hundred were European. After more than three hundred years of war, enslavement, epidemics, displacement, and maneuverings the descendants of European colonists finally gained hegemonic control east of the Mississippi by 1820. The displacement of Native Americans from eastern North America was neither fast nor easy, and the complex history of those forgotten centuries was vital to understanding emerging U.S. Indian Policy. The tactics, rhetoric, and justification used during early conflicts like the Pequot (1636-1638) and the Yamasee Wars (1715-1717) would lay the foundation for an emerging American identity, and the total war policy that devastated native populations.

For more info, and a great introduction to the history of the Eastern U.S., check out Facing East From Indian Country by Richter.