I know that some of the land was acquired by conquest, but it seems as though if the settlers showed up and started killing the Natives and claiming land, the Natives would have rose up early and there would be a lot of stories about the battles with the Natives. Also, there wasn't a lot of people that could be carried on the early trips, let alone weapons.
Anther question is how were the property divisions assigned. Like the French had a huge amount of land that they ended up giving to the Spanish and English. Why didn't the French just say they held everything to the West Coast and jack up the price to the Americans?
In "1491: New Revelations of the Americas Before Columbus", Charles Mann argues that European infectious diseases were by far the most significant factor in the European conquest of the Americas.
He disputes that European technology wasn't actually overwhelming and that in many ways, Native American technology was superior. For example, bows had better range, fire rate, and accuracy than the muskets and pistols in use at the time.
Instead Mann says that modern scholarship suggests that the Americas were much more densely populated than long thought, and that by the time Europeans arrived, the region had already been depopulated by disease. This left indigenous society in a kind of post apocalyptic nightmare where they were still reeling from a massive, unprecedented epidemic, whole towns wiped out, civil society in some places collapsed, local balance of power destabilized, and in the midst of all that enter Europeans.
Another answer to your question about why the Natives didn't rise up early when colonists claimed land, is that they sometimes did. I mostly just know about English colonies, sorry, but in Jamestown, the first permanent English colony in the New World, there were three wars fought between the English and the natives, called the Anglo-Powhatan Wars, before the natives gave up and moved away from the area around Jamestown that the English claimed. There were definitely similar battles fought when the French and Spanish settled the New World as well. You're right, it was hard to transport a lot of people across the ocean to the New World, and sometimes a lot of Europeans would die in those battles along with the native Americans, but in Jamestown at least, the English were successful enough to keep living there and keep being able to call it their land.