Short answers, pulled from the story.
The Partial Nuclear Test Ban Treaty, signed on the 5th of August 1963, banned all test detonations of nuclear weapons except those conducted underground. It prohibited tests in the atmosphere, in outer space, and underwater.
The governments of the Soviet Union, the United Kingdom, and the United States signed the treaty in Moscow on the 5th of August 1963. It formally entered into force on the 10th of October 1963, and 123 other states subsequently became parties to it.
Underground tests were excluded because scientists and negotiators could not agree on verification methods. US experts concluded that the Geneva System could only reliably detect underground detonations above 20 kilotons, and research showed that a technique called decoupling could mask a 300-kiloton blast to appear as a 1-kiloton event in seismic readings.
Castle Bravo was a US thermonuclear test at Bikini Atoll in 1954 that produced a yield of 15 megatons, more than double the expected yield. Radioactive particles spread over more than 11,000 square kilometers, sickening Japanese fishermen aboard the Lucky Dragon. The event galvanized public opposition to atmospheric testing and redirected international debate on nuclear policy.
No. France and China did not sign the treaty. France continued atmospheric testing until 1974 and China until 1980, years after the treaty came into force.
The October 1962 Cuban Missile Crisis shocked both Kennedy and Khrushchev into seeking faster rapprochement. Khrushchev wrote to Kennedy on the 30th of October 1962 proposing a broad set of agreements including a halt to all nuclear testing. Khrushchev later recounted that he viewed test-ban negotiations after Cuba as the prime venue for easing tensions between the superpowers.