Nisei (American of Japanese heritage) were regularly deployed to the European theater out of a fear that the troops would turn against the U.S. if deployed in the Pacific. While National Guard units were often kept together, everything else would routinely be pulled from all over the country.
No, perhaps to our detriment. In the German army of the same time, men in an individual infantry division were drawn from the same military area (wehrkreis). There were something like twenty of these in Germany, so they would be quite small areas. Think: recruiting men from greater Cleveland, Pittsburgh, Cincinnati, etc. This system may have broken down at some point during the war, but that was the ideal. The US Army dispersed draftees and wartime enlisted among units without any geographical basis. This, along with the combat replacement system, may have led to the bewilderment and isolation of new men.
To some degree. The 45th infantry division was made up of people mainly from the Southwest, for example. I just read that in Ian Kershaw's The Liberator.
I have never heard this claim that Japanese-Americans were sent to Europe because the military feared they might "turn against the U.S.," and I suspect it isn't true. Wouldn't it be more likely that J-A troops were sent to Europe because of the obvious confusions that might have happened with other American troops in the Pacific mistaking them for the Japanese, especially in the heat of battle? This is something that could not have reasonably happened in Europe.
And there were not very many J-A troops in the first place, so sending them to a particular place may simply have been at least partially an accident of war and a matter of where they happened to be needed most at that time. For one thing, the main emphasis was always on Europe. So, new troops -- send them to Europe. In other words, race may have played only a little role or no role at all.
Finally, my father-in-law was a J-A soldier in the war, and he was sent to the Pacific after spending some time taking Japanese language refresher courses. So, it is not even true that J-A troops 'were' all sent to Europe.