What are some of the main differences in attitudes towards immigration in the US, which defines itself as a nation of immigrants, and Sweden, which has been culturally homogenous for most of its history?

I think it's hard to speak of attitudes in those terms. Despite the US's self-image as a "nation of immigrants," the first real federal immigration laws from the late 19th century were restrictive, culminating in the quota act of 1924. I also think that the image of a culturally homogeneous Sweden doesn't reflect historical reality. In the early modern era, for example, the cultural and political gap between peasants and aristocrats was unbridgeable, which led to number of peasant rebellions. In the modern era, people from the working class were considered by economic and political elites as incapable of political participation because they lacked respectability, reason, and civility. That's why we need history, because complex facts like these don't appear in attitude surveys.
With that caveat about the limits of attitude surveys, a recent article of mine looks at immigration practices and perceptions in Sweden: " Swedish Immigration Is Not out of Control -- It's Actually Getting More Restrictive." 

Your research looks at the "immigrant question" during the Cold War and the War on Terror. In both those periods, "the enemy" was an ideology and an immigrant /asylum seeker might subscribe to that ideology or might be fleeing from it -- but what's different in the current era?