How did the nature of the Cold War change after the death of Joseph Stalin? Discuss two of the events that occurred in the 1950s and 1960s that added to Cold War tensions.

For a few years after Joseph Stalin's death in 1953, there was uncertainty about who would become the next leader of the Soviet Union. The Soviet government consisted of two parallel structures: 1) the state and 2) the Communist Party. This uncertainty caused anxiety in the West. By 1955–56, after his contenders were purged or demoted, it had become clear that Nikita Khrushchev was in charge.
On February 24–25, 1956, at a closed session of the Twentieth Congress of the Communist Party, Khrushchev delivered the so-called Secret Speech. The text was obtained by Western intelligence agencies, translated by Russian speakers in West Germany working for the C.I.A., and widely disseminated. In the speech, Khrushchev denounced Stalin's cult of personality and accused him of breaching Party principles, Party democracy, and revolutionary legality. Thereafter, he pursued a policy of cautious de-Stalinization, and there was some cause for the West to feel optimistic.
In foreign policy, Khrushchev pursued a policy of peaceful coexistence with the United States and the capitalist West. Some historians prefer the term "competitive coexistence," as competition never stopped. Khrushchev promised to overtake the US in per capita output of butter, milk, and meat. In 1957, the Soviet Union shocked the world by successfully launching the first man-made satellite, called Sputnik, into orbit. Overall, in the late 1950s, there was an easing of tensions. Khrushchev even visited the United States in 1959.
Then in 1960, the Soviet Union shot down an American U-2 spy plane during an overflight. In response, Khrushchev pulled out of a planned conference in Paris with President Dwight Eisenhower. The Soviet Union increased military spending and tensions mounted.
In 1962, the Soviet Union sought to bring nuclear missiles into Cuba. The goal was strategic parity. Khrushchev knew that the United States had secretly placed missiles in Turkey which could reach the Soviet Union and he sought to level the playing field. Meanwhile, Cuban leader Fidel Castro welcomed the defense against the US, which had tried to overthrow him with the 1960 Bay of Pigs invasion. The John F. Kenndey administration sniffed out the Soviet plan, resulting in the Cuban Missile Crisis in October 1962. Kennedy decided to establish a naval blockade of Cuba, euphemistically called a quarantine, and Khrushchev ultimately backed down. He agreed to abandon the plan, although there was a secret agreement made by John and Robert Kennedy to remove American missiles from Turkey as well.
The shooting down of the American U-2 over the Soviet Union and the Cuban Missile Crisis were two of the tensest, most ominous, moments of the Cold War. During the Cuban Missile Crisis, many people on the eastern seaboard of the United States imagined that nuclear missiles could rain down on them at any moment.
So what changed from Stalin's time? On the one hand, there was a de-escalation of tensions in terms of foreign policy, with the mutual understanding that the two sides in the Cold War could coexist. On the other, there was intense competition and, most importantly, a nuclear arms race, which threatened to wipe out civilization.

Stalin, while hostile to the West, did not take steps to antagonize the West to the point of war. Nikita Khrushchev, his successor, had to appear strong in the Communist party. He was more willing to take chances against antagonizing the West in order not to appear soft.
One event that took place during this era was the launching of Sputnik. The Soviets were able to capitalize on captured German scientists after the war and they created their own space program from German rocket technology. Many in the United States feared an attack from space; this launched the "Space Race" between the United States and the Soviet Union.
Another event was the Cuban Missile Crisis. Khrushchev literally embraced Fidel Castro and his revolution in Cuba, and Castro appealed to the Soviet Union after the Bay of Pigs fiasco, which was an attempt by the US to overthrow him. The CIA launched spy planes over Cuba and detected nuclear missiles there which could threaten the entire East Coast. The United States, under President Kennedy, blockaded the island until negotiations between Kennedy and Khrushchev led to the removal of the missiles in return for the United States removing its missiles from Turkey. In the United States, missile removal from Cuba was the headline from the negotiation as it made Kennedy look strong. In the Soviet Union, the United States's removal of its missiles was the big story in order to make Khrushchev appear the winner.
A third event was the downing of an American spy plane over Russia during the Eisenhower administration. Eisenhower directly stated that his country was not spying on the Soviet Union. Khrushchev went public with the wreckage of the plane as well as the captured pilot, Gary Powers. This was right before a peace summit between the two nations. Powers was ultimately exchanged for a captured Soviet spy in 1962.
https://www.history.com/this-day-in-history/american-u-2-spy-plane-shot-down