How did Americans justify dispossession of Indians and how did they attempt to avoid dispossession by adapting their culture to that of the invaders?
This is an essay prompt, but I am not asking anyone to write my essay for me. So far, I've used Andrew Jackson's removal of Cherokees, Thomas Jefferson's Empire of Liberty, and Manifest Destiny to answer the question.
As to the second part of your question, you can see Indians trying to adapt their culture to that of the whites throughout US history.
At times, they did this to avoid dispossession, as you say. The best known example of this would have been when the "Five Civilized Tribes" of the Southeast became fairly settled and agrarian. They had a culture with mills and even slaves.
Sometimes, they did this because they wanted to exploit the whites and use their resources. The whites were generally powerful and had material goods. By aligning themselves with whites, various Indian tribes hoped to benefit from these things.
The motivation was often plain economic greed, for land, for resources, for overland routes of settlement and transportation. To justify this greed, a rather complex social system of racism against Indians was created and maintained, even encouraged by the government and society at large. At the time of the Trail of Tears you mention, that racism against Natives was widespread and even typical in the views of most Americans. So while the Supreme Court ruled the removal unconstitutional, Jackson could afford to ignore them because removal was so popular.
The other obvious justification besides outright racism was religion. The idea that Natives were savages because they were not Christian, and because they worshipped many gods was written and spoken of widely in those days. Killing and relocating heathens was not seen as an evil by many people. Most Americans by the 1830's and 40's believed in the idea of Manifest Destiny, that God had blessed the white race with power and wealth, and therefore it was His will that we expand as a nation from coast to coast - through the Indian tribes if we had to - violently, and because of these justifications, largely guilt free.