Part 1/2
To answer this question it is necessary to delve into some context and differentiate between a couple of levels of answers. There is a tangible connection between how the "conquering of the American West" was imagined by the Nazis, how they understood American Manifest Destiny and how they conceived their own ideological vision of German living space in the East. While this is important for several concepts I will expand upon below, it also needs to be said that while this connection is there, there is also no direct connection in the sense that Nazi officials sat down to consciously study American history to find solutions to problems they faced in the execution of their own genocide as far as our sources can show us.
A lot of recent scholarship has taken up the question if and how the Nazi project of conquering Lebensraum in the European East – which entails military conquest, ethnic cleansing, German settlement, and the Holocaust – fit into the history of Western Imperialism and Colonialism, including Germany's own history of colonialism in Africa as well as the United States. Mark Mazower's Hitler's Empire, Wendy Lower's Nazi Empire Building and the Holocaust in Ukraine or in the scope of this debate Edward B. Westerman's Hitler's Ostkrieg and the Indian Wars. Comparing Genocide and Conquest have been major contributions to this debate.
As a particular subset of this debate, there have been those who have attempted to draw a direct link between the Nazis' Holocaust and a variety of other historical genocides. In Germany, Jürgen Zimmerer has developed the thesis that the Nazis were directly influenced and inspired by the German colonial genocide in Namibia in the beginning of the 20th century. Ward Chruchill (whom my friend /u/snapshot52 can tell you more about) has asserted that Hitler built his concept of Lebensraum "directly upon the US practices against American Indians". While these works provide interesting ideas for the overall debates, the problem is that in both Zimmerer's and Churchill's case the assertion of a direct influence – meaning the idea that the Nazis consciously studied both cases as preparation and inspiration – rests on very little empirical evidence. Both rely on a number of quotes from Hitler who at several occasions referred to the Volga as the Nazis' "Mississippi", the Ukraine as Germany's "India", and the people of the Soviet Union as "Red Skins".
What needs to be taken into account with these quotes, mostly drawn from Bormann's notes, is that they rest not upon a conscious and ardent actual study of the American genocide, the British colonial policy in India etc. but rather on Hitler's and the Nazis' pop-cultural and limited understanding of these historical events. As Edward B. Westerman writes in his above mentioned book, which provides a meticulous comparison of similarities and differences between the Ostkrieg and the Indian Wars and underlying structures, regarding such a comparison made by Hitler in one of his table talks:
The importance of Hitler's declaration [that the American West was conquered by "the white man shooting down the millions of redskins to a few hundred thousand"] is not related to the accuracy or inaccuracy of his assessment, but rather to his own belief in the historical truth of this statement. Indeed, if Hitler envisioned the conquest of the American West as a process that included the murder of millions, then he could also imagine the subjugation of German Lebensraum in the East thorugh a process of mass annihilation. Hitler's boast on creating a new frontier in the East to rival that of the American West in the previous century not only demonstrated his limited understanding of US history but, more importantly, exposed the genocidal fantasy that framed his conception of conquest and exploitation in Eastern Europe.
What Westerman asserts here is that it is not so much the concrete and direct influence, meaning the study of the particular historical case, that influenced the Nazis but rather underlying structures and general methods that the case of the American genocide and others had already established when it came to colonial and imperial actions.
This idea of a colonial/imperial framework to Nazi atrocities is older than one would assume and predates current discussions of postcolonial approaches in historical scholarship, though the latter have certainly helped invigorate this approach again. The idea to approach Nazi and German actions during WWII from this angle emerged from Hannah Arendt's book The Origins of Totalitarianism, which while otherwise form a historian's viewpoint rather useless, posited that colonialism and imperialism had as large European phenomena been the "greenhouse" of the totalitarian state in that it created the bureaucracy of massacre, the intertwined relation between state administration, violence, and exploitation based on the idea of re-ordering humanity into "master and slave races". What Arendt essentially says is that the idea of creating a state administration with the aim of exploitation and violence against people considered inferior and destined to be exploited, is a trans-national phenomenon that originated in colonialism and imperialism.
Recently, this idea was taken up by Enzo Traverso in his book The Origins of Nazi Violence where he writes that both the völkisch ideology of Germany in its anti-Semitism and the colonial discourse of Europe at the time contributed to an aggressive, non-egalitarian, antidemocratic nationalism geared towards conquest. Colonialism and Imperialism as practices solidified the view of the world that stated that a people had the right and mission to use massive violence for the purpose of subjugating others and in service of their exploitation. Hence, writes Traverso: "In National Socialism we see the coming together and uniting of two paradigmatic figures: the Jew, the "others of the Western world and the "subhuman", the "other of the colonial world."" Referencing Arendt, Traverso describes colonialism as the synthesis of massacre and bureaucracy with modern racism justified by Western science and bureaucracy as manifestation of Western rationalism coming together to create the massive violence of colonial and imperial enterprises and finding their pinnacle in the Nazi expansion to the "living space" of the East and in the terror of camps designated to kill those viewed as dangerous and subhuman with industrial production methods.
Another analytical approach that has been added recently and that stems from the postcolonial approach to history in general is that of the "colonial archive". What it means is that over the course of European colonialism and imperialism, a shared discourse of knowledge between the different colonial powers originated that concerned the treatment of colonial peoples and the various techniques of violent rule in European colonies. Such research is f.ex. exemplified in the various works on the Philippines where it is highlighted that 26 of the 30 American generals there had experience with oppressing Native Americans and often directly referenced this during their time fighting in there. This approach understands Nazi Germany as part of this colonial archives and integrated the people of Eastern Europe into this matrix of knowledge in the role of the "inferior native".
The advantages of this approach as exemplified by the above mentioned books by Mazower, Lower, and a couple of others is that from an analytical standpoint, understanding the Third Reich and its new order as an empire in the same sense of Western Empires of the 19th century helps conceptualize the very strong relation between occupation, exploitation, Holocaust, and administration of the various territories. Within this analytical approach of understanding, connections and lines can be drawn between plans to "Germanize" Western Poland, the Holocaust, the Hunger Plan, intended to decimate the population of the Soviet Union, the cultural and educational policies of the Germans in Poland and the USSR such as they were and a lot more. It helps us better find a way to understand how the New Order could be legitimized in the eyes of those carrying it out. What I mean by the last point can be observed f.ex. in how German bureaucrats and administrators in Poland understood their own role: They were part of an effort to "restore a natural order" disrupted by Jewish influence in which the incapability of the Poles to run a state or be more than a people subservient to their German overlord was restored. This can be tied directly to mechanism of European colonialism and imperialism where indigenous and native polities were torn down with the ideological idea of the Europeans being more "advanced" and "developed" and only working to install an order that was "natural", meaning one in which those who were not as "advanced" were subservient.