Answering on mobile so I can't give proper citations but I'll try to point you in the right direction.

Klaus Fuchs worked inside the Manhattan project on the plutonium implosion problem. He passed technical information to the Soviets both on the atomic and hydrogen bomb programs. He confessed to these crimes in 1950 and his conviction lead, eventually, to the Rosenberg trials.

He probably didn't contribute much on the hydrogen bomb problem as real progress on fusion post-dated his activity period but his intimate knowledge of how implosion worked, to say nothing of other key information including the gas centrifuge process.

Wikipedia here, not a real citation but the bibliography is good: https://en.wikipedia.org/wiki/Klaus_Fuchs

It depends on what you mean by "the science." The type of information that was conveyed had to be of the sort that could be written down by individuals and passed on. So a few reports. A few interviews. A few things that looked like blueprints. Some equations. In one instance, a small sample of material (from Canada). But you have to remember that in the 1940s, conveying a lot of information was hard to do, especially clandestinely. And to make an atomic bomb requires produced millions of pages of reports, experiments, and so on.

So the Soviets had access to some of that. They did not have access to most of that. They had to replicate the science on their own end with an effort that was the same size, more or less, as the Manhattan Project. This was not incidental: to enrich uranium, to breed plutonium, to actually assemble a workable bomb — this all requires much more than information that can be easily or explicitly transmitted. It requires not only developing a massive knowledge base of the kind of stuff that can be written down ("explicit knowledge"), but also the kinds of hands-on, "know-how" that comes from actual experience in building the facilities and components ("tacit knowledge"), the latter of which cannot be transmitted easily by spies.

To give some examples, here are Soviet drawings of the Trinity/Nagasaki device based on intelligence data, and here is data they got on an early hydrogen bomb design. The quality is good for both of these, no doubt because the spy passing on a lot of the most important data, Klaus Fuchs, was an unusually competent physicist who also had a photographic memory. But the physics and designs are only one component of making a bomb — there is also chemistry, metallurgy, engineering, ordnance... things that Fuchs and the other physicist spies had a harder time passing on because they were not experts in these fields and did not often have access to the work being done at other sites.

For what the Soviets got, they got good information, but there was so much that they did not, and could not, get.

As for Stalin himself, I don't think he looked at any of this stuff himself, directly. It was funneled to Igor Kurchatov, the top scientist on the Soviet bomb program, and he briefed Stalin on their work.