Ugh. Should have left that part out, and now, I spent too much time trying to give you a good response. ;)

Japan was hedging it's bets at the top, giving the option to explore a soviet mediated conditional surrender, ie. the maintenance of their post ww1 gains, maintenance of government, etc. They, Togo and Suzuki, had been considering it for a while. However, it wasn't well recieved by the mil, who thought it at best a tactic to garner an ally and to delay for time to make a defense or to get a new opening. And that is key, as it's the military had the majority of spots and overall control of decisions, only breaking after the initial bombing and soviet involvement (to a degree the soviet involvement might have been more shocking for the mil, as it wasn't known if the US had anymore bombs). And a military coup to continue the war was a chief concern, so much so that any discussion of it was done in secret. This is also likely the reason for their screwups on not responding to/inability to negotiate, as once it was made known, a coup could happen. The disconnect between the cabinet levels pragmatism and the lowers zeal didn't help any outside impression, and is one reason they had both a botched coup and yet an orderly surrender.

The soviets disregarded the request and invaded soon after it was made. This removed Japan's possibility of using the Soviets as a negotiating party/leverage with US/GB/China. This likely means a combination of soviet views/reasons:

a. the soviets thought their unconditional surrender was imminent and needed to grab land quickly
b. They didn't think a unconditional surrender was imminent but needed to get on with their agreement with the others (They had to go to war on August 9th as that was the yalta agreement, 3 months after may 9th, and it's not like the sped up thinking they were close to surrender and needed to get in quickly)
c. That the mullings of conditional surrender was a delay tactic to regroup/dig in and so should be ignored
d. Some combination of US/GB/China would or could not entertain a conditional surrender and thus it was moot, potsdam for instance
e. Japan was just looking for a new partner to help with US/GB/China and it would be not kosher to entertain it as they already had Yalta/disagreements in Europe/fights over the germans surrender
f. Japan was weak and offered nothing, especially consider russo Japanese history

But consensus is just what's being carried out, not the state of all opinions. It's the reason I included "could". For example, the US may have thought that it was a strong possibility they may not surrender to any reactions, but tried anyway by dropping leaflets and more. Japans offers for a soviet negotiated conditional surrender were known to the other allies as they were decoded. Now, you could go down the path of if the US/GB/China should have gone the route of entertaining a conditional surrender to avoid continuing. But conditional surrenders can drag on for years, the war continuing all the while, and used to delay for an opening or a peacetime regrouping. It's why at all the conferences they insisted on unconditional.

Some the cynical attitude towards it was that japan wanted to play the cost game near the end, where they held that they'd fight vigorously unless they got good terms or something changed. And it's not a terrible plan mind you, as for example, the US has shown weakness to such costs in subsequent years, though for less popular wars. Or that Great Britain for example, despite being alone and under imminent invasion, saw the entrance of the US and the Soviet Union into the war. Some of this is also likely influenced by the potsdam terms, indicated the allies in the pacific did not seek Japans complete destruction. Some of this was to make surrender look attractive but it also betrays a weak stomach (despite the firebombings indicating otherwise). It only appears that threat turned into a real thought post atomic bombings (per their comments in the meetings about the bombs).

It's betrays the delusion of were they thought the war stood, that the atomic bombs meant anything changed or had even considered the USSR as a negotiator. They had already been firebombed, and thus looked at possible annihilation without the atomic bombs, they had no allies, the soviets had given warning that they were not on their side by not renewing the non aggression pact in april though they did not weigh it like they should have. Sorta the last desperate and illogical actions of a flailing government. It's one reason why the whole episode is confounding, especially later, as a "rational" actor would have surrendered earlier.

Per your point on the soviet invasions.
The soviets got bogged down on both landings quickly and were helped by the surrender of a large portion of the japanese forces starting on the 15th and still taking till well after the surrender to take sakhalin and kuril. This likely would have happened to hakkido, if they weren't repulsed (doubtful they'd be fully dislodged, but overextended supplies are bad when getting bogged down). I'd have to dive more into the planning on hakkido to see the state of it, as Sakhalin/kuril are often seen as quick risky land grabs as they were launched after the bombings/invasion of Manchuria, and surrender on the 15th. Still the Soviet style of pressing might have made some gains, though it would have been limited to Hokkido. For your original post, it depends on if Japan thought Soviet occupation of Hakkido as a breaking point to take an unconditional surrender, (Okinawa comes to mind as a failed breaking point, though it did seem to set some stuff in motion), but I doubt it as, as it's always been sorta a backwater for Japan.