To address the second part of your question first, the Allies had prepared Operation Downfall, an amphibious invasion of mainland Japan by American and British forces that would've been the largest amphibious operation ever carried out had it taken place (surpassing even D-Day). It involved using Okinawa as a staging point for an invasion of Kyushu in late 1945 (Operation Olympic), with the intention of using the island's air bases to provide support for an invasion of the Kanto Plain near Tokyo in 1946 (Operation Coronet).
Obviously this never took place since the Japanese surrendered, which ties into the first part of your question and the assumption it makes: that Japan surrendered primarily because of the atomic bombings of Hiroshima and Nagasaki. Ever since the Allies gained air and naval supremacy over the Japanese, they had been carrying out widescale firebombing operations in major Japanese cities. These attacks levelled entire cities, killing 100,000s and displacing millions; a problem exacerbated by the fact that Japanese cities were constructed out of more wood than European ones at the time, which meant that these fires often burnt uncontrolled for days. What this means is that by the time the nuclear bombs were dropped, it wasn't exacting shocking to the Japanese consciousness that the Allies had levelled two cities full of civilians.
What many people don't realise is that the Little Boy bomb was dropped on Hiroshima on August 6th, and Fat Man wasn't dropped on Nagasaki until August 9th. Why is that important? Because a), it shows that at least the first bomb didn't immediately push the Japanese to surrender, and b) it coincides with the Soviet invasion of Manchuria (a Japanese puppet state in North Eastern China on the Soviet border) which took place on the same day. The Japanese army was resource starved (sound familiar?) and had adopted the policy of pushing into the resource rich islands in the south and continuing their war in mainland China, and wasn't really interested in prosecuting a war with the Soviets in the middle of Siberia. The Japanese, despite being formally allied to Nazi Germany had signed a non-aggression pact with the Soviets in April 1941 after an undeclared border conflict (the Battles of Khalkhin Gol) on the Mongolian-Manchurian border (Soviet and Japanese puppet states, respectively). The battles resulted in a decisive show of Red Army superiority over the Imperial Army and only reinforced opposition to the Japanese invading Siberia. It's worth noting that this pact was signed only two months before Operation Barbarossa and the outbreak of war on the Eastern Front. Not long before the Soviets broke the pact (which was set to last until 1946), there were Japanese diplomats in Moscow trying to keep the Soviets from getting involved. But as the war in Europe wound down, it was just a matter of time.
All this is to say that the Soviet Union breaking their non-aggression pact is an underconsidered factor in most people's minds when it comes to the surrender of Japan and the end of WW2. Certainly the bombs were critical. However their importance stemmed largely from the idea that these weapons of mass destruction could be mass produced on a level that would make war apocalyptic in nature,and not necessarily from the destruction they actually caused, as Japan had already seen so much comparable destruction by this late stage in the war. This is further evidenced by the fact that scientists working on the Manhattan project had feared as much, and immediately after the fall of Japan the Allies were working to prevent the Soviets from getting the bomb.
As far as other bombs are concerned, at the time of the surrender the Allies had no more bombs. The Thin Man bomb (a different design for a fourth bomb) was abandoned in 1944, and you know the stories of the other three. Gadget Man was dropped in the Trinity Test, the first ever detonation of a nuclear weapon, and the last two were dropped on the cities of Hiroshima and Nagasaki.
In short, the Allies were prepared to invade Japan if that's what it took, however the Soviets breaking the non-aggression pact and invading Manchuria was the straw the broke the camel's back. There were some in leadership positions who wanted to continue the war until the bitter end even in spite of the Soviet declaration of war. This culminated in the Kyūjō Incident on August 14-15 1945, where the night before Japan announced their surrender (not to be confused with the official signing of the surrender which took place a fortnight later on September 2), officers from the Ministry of the Army and Imperial Guard attempted to place the Emperor under house arrest and keep the army fighting. Of course this failed, those involved were killed, and the surrender was announced the next day.
The day after the Nagasaki attack, and he had been informed that the next bomb would be ready by around August 17-18 (a week), Truman ordered that no further weapons would be used until his order.
However the assumption from within the military was that if Japan did not accept unconditional surrender soon, it would be likely that atomic bombing would resume. The questions at hand were, if that was the case, what would the best approach be? There were those who advocated the bombing of Tokyo, and Truman himself, in a discussion with British officials on August 14th (shortly before they got news of the surrender), indicated ruefully that he might feel he had to do that. There would have been a lot of pressure on him to do something like that — various officials were planning to lobby him with this idea, both high members of the military and high civilian officials.
The other idea that was being considered, largely by Chief of Staff Marshall, was that they might "save up" several weapons and either use them at the same time (e.g. three cities at once) or to use them later in the service of a land invasion. The Manhattan Project production system at the time could produce a little more than 3 bombs per month operating at full-effort, and potentially even more if more radical bomb designs were undertaken that would help "stretch" the bomb fuel further (they were contemplated at the time, but not pursued — but probably would have been looked into vigorously had the war went on).
In any event, no final decisions were made, or had to be made, either way. So we do not know what would have really happened; it ultimately would come down, in this instance, to Truman's own individual decision, and such things are hard to predict.
As for invasions: the invasion of Kyushu had been approved by Truman in June 1945, and was not scheduled to begin until November 1, 1945. So there was still considerable time before that began. Truman himself does not seem to have assumed it would be necessary after the Soviet Union got involved in the war, and the general feeling in Washington was that certainly the combination of atomic bombing and the Soviet invasion would cause Japan to surrender quickly. I do not get the sense that anyone in Washington actually thought an American invasion would be necessary at this point.
Truman pointedly did not approve of the invasion of the main Japanese island of Honshu in June 1945; he took a "wait and see" approach, on the assumption that it might not be necessary. For whatever reason, this gets overlooked quite frequently — I think it is telling about his mindset.