As the title says, would an American citizen have been able to obtain a tourist visa to tour the Soviet Union during the Cold War, particularly post-Cuban Missile Crisis? If so, what would the process entail, and could they expect some serious questioning upon returning to the United States?
Well, prior to the October 1962 crisis, Lee Harvey Oswald didn't seem to experience much difficulty. His travel to the Soviet Union in October 1959 is quite well documented. He travelled to Europe, via Britain, then flew to Helsinki where he was granted a one week visa to travel in the Soviet Union.
He subsequently overstayed the visa, obtained permission to stay in the Soviet Union and married. He returned to the U.S. in June, 1962 with his wife and infant.