What were Russia's ideas/views on JFK's assassination? Did anything interesting come out when USSR fell? What about Cuba?
Yes. One of the most interesting bits that came out deals with conspiracy theories, or rather their origins.
The Mitrokhin Archives, a treasure trove of confidential KGB information that was brought over by the KGB's official archivist when he defected in 1992, explicitly explains how the KGB itself was the originator of the conspiracy theory that the FBI or a nondescript U.S. government cabal or people with strong business interests were responsible for assassinating JFK.
Here's what's in the book:
In September 1961 the CPSU Central Committee allocated another 55,000 dollars for the next two years to allow Marzani to expand his publications. He was given a further 10,000 dollars a year to cover advertising costs. When the young KGB officer Oleg Kalugin, stationed in New York in the early 1960s under cover as a Radio Moscow reporter, paid his first visit to one of Marzani’s receptions, he found his apartment “filled with a motley assortment of Communists, liberals, and KGB spooks-all of them watched, undoubtedly, by FBI informers in attendance.”
Among the books published by Marzani in 1964 was the first volume on the Kennedy assassination to appear in the United States, Oswald: Assassin Fall-Guy? by the German writer Joachim Joesten. At the beginning of the book Joesten expresses his “heartfelt thanks . . . to Carl Marzani, a shrewd and hard-hitting publisher in the finest American tradition, who put his whole heart and soul in this book;” Marzani succeeded in publishing it within five weeks of receiving the manuscript. Joesten supported Moscow’s line in pinning the blame for the assassination on a conspiracy by right-wing racists, chief among them “oil magnate H. L. Hunt:”
The KGB is responsible for some of the most pervasive "classic" conspiracy theories. They even invented the one about the U.S. faking the moon landings.
Source: cited bit is from "The Sword and the Shield: The Mitrokhin Archive and the Secret History of the KGB" by Andrew and Mitrokhin.