For George Papadopoulos, the path to politics—and to President Donald Trump —was a circuitous one. The former Trump aide, who pleaded guilty in early October to lying to the Federal Bureau of Investigation about his contacts with Russians during the election campaign, entered the political world in 2015 with no campaign experience and little foreign policy background. A onetime oil and gas consultant who then served as a research associate at the conservative Hudson Institute and touted his Model United Nations experience as a credential, Mr. Papadopoulos began the 2016 presidential campaign by working for Republican Ben Carson. He was...