This question comes up far too often. Here is a recent instance of the "what ended the Pacific War?" debate.
First, lets pick apart your question, as you're really asking three separate things:
Question 1 has been answered pretty conclusively by people working on Japanese sources as "probably not." It seems that the Japanese leadership cared more about the wrench that Soviet entry into the war threw into their plans for a negotiated settlement. That does not mean that the atomic bombs did not play a role. It just means that they were one of several factors, and other factors may well have featured more prominently in the minds of decision makers.
The Japanese Leadership had been pursuing Soviet mediation in a surrender, against their ambassador in Moscow's advice, all summer of 1945. Knowing that the war was already lost, the Supreme Council for the Direction of the War had pinned their hopes on a favorable surrender that preserved the state structure and the position of the Emperor on the (objectively unlikely) hope that the Soviets could be convinced to act as mediators.
On Aug. 7, after the Hiroshima bomb, Prime minister Tōgō telegrammed Ambassador Satō in Moscow, stating that the situation was getting desperate and that "We must know the Soviet's attitude immediately" (Hasegawa, Racing the Enemy, 185).
News of the Soviet declaration of war and the Nagasaki bombing both arrived on August 9th. Emperor Hirohito learned of the Soviet entry to the war and summoned Lord keeper of the Privy Seal Kido Kōichi at 9:55 am, telling him "The Soviet Union declared war against us, and entered into a state of war as of today. Because of this it is necessary to study and decide on the termination of the war," according to Kido's Journal (Hasegawa, 198).
Shortly after, they convened the War Council. At that meeting, they learned of the Bombing of Nagasaki. According to the official history of the Imperial General Headquarters, "There is no record in other materials that treated the effect [of the Nagasaki Bomb] seriously." Similarly, neither the Foreign Minister nor the Chief of the Navy General staff mentioned Nagasaki in their memoirs of the meeting (Hasegawa, 204). The hawkish members of the council eventually fell to the peace faction and agreed to accept the Potsdam Proclamation, but still debated a 1 condition surrender vs 4 condition surrender.
Kido then met with the Emperor, and afterward the Emperor agreed to call an imperial conference, at which he supported Foreign Minister Tōgō's proposal, saying "My opinion is the same as what the Foreign minister said." All the members, including the war party signed the document in the early hours of August 10th (Hasegawa, 213). With that, the basic outline of surrendering was complete. Although the Emperor was the deciding "vote," the deliberations show that his decision was shaped by those who were convinced to surrender by Soviet declaration of war to put his weight behind the plan they had laid out.
In the intervening days between the 10th and the final surrender on the 15th things were fairly chaotic, including an attempted cop by more radical members of the Army General Staff. The US rejected the conditional acceptance, and the army thought it was an unacceptable violation the national polity and the emperor's place therein, leading to a stalemate in the leadership. Again convinced by Kido, the Emperor convened a combined conference of the Supreme War council and the Cabinet in order to impose his decision for unconditional surrender on the 14th.
The Emperor's speech announcing surrender was broadcast on August 15th (planned for back on the 11th). It cited the bombs as a reason for surrender, but that is not sufficient evidence to determine the reasons for ending the war. First, that speech was carefully prepared and edited for public, military, and American consumption. Second, it is only one of several sources. Of the contemporary sources on why Japan surrendered, 3 (Konoe on Aug.9th, PM Suzuki's statements to his doctor on Aug. 13th, and Hirohito's Imperial Rescript to Soldiers and Officers on Aug. 17th) speak exclusively about the Soviets, 2 (Hirohito's Imperial Rescript on the 15th and Suzuki's statements at the cabinet meeting of Aug. 13th) speak exclusively about the bombs, and 7 speak of both (Hasegawa, 297-298). Obviously both played a role, but a close examination of the process of decision making gives the deciding edge to Soviet entry.
However, Question 2 has also been answered more or less in the negative. There is no way Truman could have known whether or not the bombs would end the war, and the US was still planning for a land invasion. He could not have known the thoughts of the Japanese leadership, but he had indications that the war would continue. In other words, US leaders hoped that dropping the bombs would end the war, and their strategic analysis suggested that it was a worthwhile strategy.
Question 3 has largely been sidestepped, since I've said they didn't know. However, that does no preclude the possibility of other motives in dropping the bomb. Sure, the war against Japan was the primary motivator, but it would be naive to imagine that US leaders didn't consider the postwar implications. They did indeed appreciate the effect it would have on the Soviets, but that was an ancillary benefit, rather than a primary goal. There were also simpler reasons, like the sheer cost of developing the bombs; putting them to use would help avoid a domestic backlash over the cost of the Manhattan project.
Well it is very difficult to argue "what if's" but there is at least some validity to this claim because by 1944-1945 the US's island hopping campaign was very successful and it was just a matter of time before the mainland was reached. That being said though I've read that the average japanese person was prepared to defend there homeland with sharpened pieces of bamboo so the war may have really dragged on for a few more years.
And the US definitely used this as an example of their force on a global scale, but, its debatable whether or not that was the main driving force behind the decision.