Irrelevant; they surrendered due to August Storm.
No they didn't. They might have surrendered due to the Manchurian Strategic Offensive Operation, which never had that name - it doesn't fit the way the Soviet Union named operations at all.
It's not at all clear, AIUI, what the tipping point was, or if there even was one. The start of the Soviet invasion of Manchuria and the Nakasaki bombing occurred on the same day, and were discussed at the same cabinet meeting. Hirohito mentioned the atomic bomb specifically in his surrender broadcast, but that doesn't necessarily prove anything.
For what it's worth, I think that the near-simultenaity of the two was a key consideration. The atomic bomb alone might not have been enough. The Soviet Union entering the war might not have been enough. Even both, but with a substantial (weeks-to-months) time between them might not have been enough. And note that even with both occurring, there was an attempted coup in favour of continuing to fight!
Allied ground forces landing in Japan would also be another thing that would weigh heavily in favour of surrender, IMO - but again, not necessarily enough on its own. As far as I know, there has never been a successful invasion of Japan, and nobody has seriously tried since the Mongols in the thirteenth century. The US doing so would probably be a major shock to Japanese confidence; I suspect a Soviet invasion wouldn't have been able to achieve a foothold, so wouldn't have the same effect. Sufficient suffering of the population probably wouldn't be enough to force surrender on its own, given the indifference that the government seemed to show, but if inflicted long enough, the ability to sustain war would disappear.
At that point, the line between 'continued blockade' and 'Japanese genocide' is paper-thin.
The soviet invasion and the two atomic bombs all were within three days of each other, you can't split them apart. the reality is it's not either/or, bomb or soviets, but both
The IJA didn't really care that much about the bombing at home certainly not the commands abroad.
The home authorities didn't really care too much about the red army in Manchuria
That's why it took Hirohito surrender addressing the nation and talking about the bombs, and then when the IJA didn't surrender after that it took a specific command (plea) to the IJA abroad to surrender that mentioned the Soviets
(and frankly even than there was an attempted coup because while the IJA talks about loyalty to the emperor it's only really when he says what they want to hear, and when ever they don't hear it why it must be because he's being misadvised by traitors and must be saved!)
In regards the threatened invasion of Japan that was actually the Japanese plan! The Japanese high command felt that if the US and UK & Co invaded they could inflict so many casualties on them that they would be forced to compromise on the unconditional surrender demands and come to the table or retreat whish would more politically unpalatable for the wallies, the whole thing possibly being facilitated by a neutral USSR
But both the atomic bombing and the USSR declaring war and them steamrolling over the Manchukuo army caused issue for that plan
1). The atomic bombing* what they knew was unlike mass bombing raids that required large numbers of planes and aircrews and specific situations to cause mass damage and death, now seemingly a single bomber could do the same. This presented them with the potential reality that unlike years of conventional bombing in Europe and months in Japan it was possible that country could now be functionally destroyed from the air, no invasion needed
2a). teh USSR was clearly not Neutral anymore so the USSR acting as impartial uncommitted facilitators was done
2b). While Japanese world views might still have allowed them to think the US and UK & Co were shy when it came to casualties, they had no such illusions about the USSR being willing to to suffer them if they became involved in any invasion. They also know that once the red army is standing on something the USSR is not going to let go of it
EDIT: there is also the point about the ongoing blockade, there was already food shortages on the main island, teh IJA probably wouldn't have cared that much. They know they will first in the food line, and had the attitude that the civilian's role was to primarily to give teh army what they needed to resist and then endure in order to eke out some kid of positive result. But once starvation starts kicking in it will kick in hard and fast
*and we have to remember the Japanese did not know how many bombs the US had, what production lines was like and so on. What they did know as was the thing their own guys had said was not feasible as a deliverable weapon was now deliverable as a weapon,