This is a complicated question with a complex answer, I'll try to make it as clear as possible.

The Soviets already knew about the possibility of nuclear weapons and had a program underway during WWII. Indeed, the Soviets had begun mining Uranium as part of this project starting in 1942. However, they were much more resource constrained than the US during the war so they put it on the back burner for the duration of the war. It's important to stress here just how exceptional the Manhattan Project was. It wasn't just one weapons development program, it was akin to running about 5 programs all at once, in parallel, something that no other economy was capable of doing and which ultimately was what enabled the program to produce functional nuclear weapons by 1945.

The Soviet program was gifted with many talented scientists and I think there's little doubt that they would have been able to produce nuclear weapons successfully on a similar timeframe to what they did manage to achieve. However, the program did exploit secrets gained from espionage quite heavily. Partly this comes down to the choices of Stalin and the Soviet leadership forcing particular design choices. You can see a parallel in the Soviet reverse engineering of the B-29 bomber (several of which had landed in Soviet territory during the war), Soviet engineers were capable of learning from the aircraft and designing a new vehicle from scratch to match its capabilities but Stalin wanted an aircraft that was as close of a copy to the B-29 as possible (which became the Tu-4).

The first Soviet atomic bomb (RDS-1) was substantially a copy of the US's "Fat Man" Plutonium implosion bomb design. So ultimately the program was immensely dependent on stolen nuclear secrets, but the question of how quickly they would have advanced on their own is still an open one. As I said it's very likely they would have succeeded regardless but it probably would have taken a bit longer.

Additionally, the use of nuclear weapons during the war did put significant pressure on the Soviet political establishment and caused them to heavily prioritize weapons development after the war. The program went from being on the back burner to having a functional Plutonium producing reactor within only 6 months, for example.

Also, the Soviets were able to learn precisely which designs were most useful in producing atomic bombs, which made it possible to significantly accelerate their development program by concentrating on the most worthwhile routes first. For example, the Manhattan project pursued several different weapons designs (gun assembly, implosion) and several different fissile bomb fuels (Plutonium, Uranium-235, and Uranium-233) as well as several different methods of procuring each, especially several different methods for enriching Uranium (electromagnetic separation, thermal diffusion, gaseous diffusion, centrifugation). Ultimately, some of these methods proved to be either dead-ends (thermal diffusion, production of U-233 from Thorium, gun-assembly with Plutonium), too inefficient (electromagnetic separation), or too technologically immature (centrifugation) to be useful. The Soviets had the benefit of knowing all of this and being able to pursue the easiest and fastest path to the bomb: implosion assembly using reactor produced Plutonium.

As a side note there's an open question of just how much the Soviet weapons program relied on captured German scientists after the war (much like the US's rocket program did). They were instrumental in certain aspects such as perfecting the centrifugation process for Uranium enrichment (which didn't affect the development of the first bombs but substantially affected the ability to mass produce bombs during the height of the Cold War), but many of the core breakthroughs still came from Soviet scientists (especially Andrei Sakharov).

Brief answers:

1. Soviet scientists knew about the possibility of nuclear weapons the same as American or European scientists. This was based on open research. They also had spies in the Anglo-American program, so they knew what was being worked on. It wasn't a surprise, at least to the few members of the Soviet hierarchy who were privy to the espionage data. Stalin wasn't surprised, in any case.

2. The Soviets had a small research program during the war that was mostly focused on sorting through espionage information and doing small studies. They did not have a larger program because they were busy fighting World War II. After Hiroshima, Stalin ordered a full crash nuclear program, and their work really began. The Soviet nuclear program was the same size, in essence, as the Manhattan Project. It was not especially fast; the Manhattan Project took 2.5 years to get a bomb after seriously trying to develop one, the Soviets took about 4 years.

3. The Soviets did have espionage information but they did not use it in the way that most people think they did. They didn't just say, "ah, now we know this," and build off of it. Why not? Because they didn't 100% trust the espionage. They instead used it as a semi-guide and a "check" on the work of their own scientists. So their program was guided along the general lines of the Manhattan Project (adapted to the Soviet context), but they re-created all of the research and even followed up on things that the espionage told them wouldn't work (like centrifuge research) because they weren't sure if the espionage was correct or that the Anglo-American program had really discovered everything (and in some cases, the Soviets found things the Manhattan Project did not). So the basic answer is, they used the secrets to augment their own research, but the line between the two is not as distinct as one might think. It isn't clear that the espionage saved them much time, if any, because the timetable of their bomb project was set by other factors (like access to uranium ore).

The best overall comparative book on the Soviet and US nuclear programs, and the use of espionage, is Michael Gordin's Red Cloud at Dawn.