One of the justifications for the use of nuclear weapons on Japan is that it was the only way to end thr war without a massive invasion of the home islands, which would have been terribly costly in human life. The implication, sometimes explicitly stated, is that Japan would never ever surrender unless totally destroyed.
But this recent post and others state that Japanese strategy from the start was to make the cost of war so great as to persuade the US to make a negotiated settlement to end the war.
By the time the invasion of the Japanese home islands was imminent, was Japan still hoping to reach some kind of peace settlement? Was a negotiated peace on the table at all for the US at this point? Did the knowledge of nukes by a tiny number of US officials impact this calculus at all?
Without the bomb, you'd still have the Soviet declaration of war and invasion of Manchuria and threatened invasion of the Japanese homeland. The Japanese military had already determined that if that happened then their defenses would fold. And then they would run the risk of ending up occupied by the Soviets, or divided in two like Germany.
So it's impossible to know, but there are pretty good reasons to think that Japan would have capitulated fairly soon even without the atomic bomb. Likely prior to the November 1st US invasion date.
By the time the invasion of the Japanese home islands was imminent, was Japan still hoping to reach some kind of peace settlement?
The US invasion of the Japanese home islands was not set to begin until November 1, 1945. So it was still a bit away when they surrendered.
When talking about "peace settlements" it is important to clarify which group of "Japan" one is talking about. The Supreme War Council was dominated by militarists, but there was a swing faction who were seeking a conditional surrender mediated by the Soviet Union. It didn't get very far because the Soviets were not interested. See this thread. So yes, some parts of the SWC were seeking a conditional surrender, but in an oblique way. The dominant militarists were not and never were. Some of them were totally delusional about their prospects.
Was a negotiated peace on the table at all for the US at this point? Did the knowledge of nukes by a tiny number of US officials impact this calculus at all?
It is possible that, without the atomic bomb to buoy him, Truman might have considered the advise of some of his advisors (notably Stimson and Churchill) to allow the major Japanese "condition" that the "peace faction" of the SWC favored — to retain the imperial office. It was clear from polls at the time that Americans didn't care about this at all, and if that had been his only option, I wouldn't be surprised if he'd have taken it. (In the end, they let the imperial office remain anyway.) But it's hard to imagine any concessions beyond that, and even that's a big "what if" — it would have been something Truman wouldn't have wanted to do, for sure, just on the principal of it.
The effect of the atomic bomb on Truman at Potsdam was him feeling that he no longer needed the Soviets to intercede (they did anyway) and that he didn't need to compromise at all with the Japanese. (I wrote a bit on this, here.)
While more can indeed be added, I covered end of the Pacific War in this answer over here, stressing the Allies' refusal to accept anything short of Unconditional Surrender as per the agreement made during the Casablanca Conference in 1943, and reiterated in the Potsdam Declaration in July of 1945. In short, the Allies were never interested in accepting a negotiated surrender, especially because by the Summer of '45, Japan was in no position to negotiate anything with the Allies, especially after the Soviet Union's declaration of war on Japan.