What interactions with Muslim nations, other than military, do Americans have ?

I'm uming you mean individual Americans, not the government. For decades, Americans have worked in the Middle East in non-military jobs - construction, health care, education, etc. They love to do it because the pay is tax-free. Although some live in company enclaves, a lot live outside compounds and make friends with ordinary people on the street. And I know some military families who have lived on permanent bases there who also became friends with the "natives". I don't know about formal stuff like sports teams, etc., except in the Olympics. I know some rock bands have performed there, although it is difficult because of the cultural restrictions. I don't think many Americans could wrap their heads around Arabic music, though.