Depends on the circumstances. There might or might not be a Bolshevik state. Given its nature and that of western Europe I'm doubtful whether the western powers would look for it to be an ally against the Germans or consider it to be a worthwhile one. [Remember that they were willing to support white groups in the early states of the civil war while still fighting the Germans.]
Of course they were willing to support the White groups. The Bolsheviks certainly weren't willing to fight the Germans...at least at that time. Whether the primary state in Russia is bolshevik or not, if it is too weak to be an useful ally, Britain wouldn'd need a Japanese alliance to watch it, and if it is strong enough to be an useful ally, it would be a more useful ally than Japan as well as a natural enemy of Japan.
Its based on more than sentiment, although that should never be underestimated. There is the fact that Japan is facing a pretty threatening US that has made clear its hostility to Japan. Having a major power as an ally and accepting it as an equal is significant. Don't forget as well that while some elements wanted outright military expansion others were looking towards commercial and economic expansion and some were looking to work with and help develop China as a fellow Asian nation.
By 1917, Japan had long stopped looking at China as an equal. By 1920, the Guandong Army and the Imperial Way faction were already entrenched in their power and influence, and determined to expand Japanese control in China. Considering the insane amounts Japan was quite willing to spend on its military long before the WNT or the abrogation of the Anglo-Japanese treaty, it seems clear that the peaceful expansion party were already well in eclipse. And of course, Britain is completely useless as an ally against the only power for which Japan could use an ally, leaving just sentiment. Though I agree that sentiment can do some crazy things, it is exceedingly doubtful that sentiment would be enough to persuade Japan to go for peace rather than militarism, when militarism had been steadily gaining in strength for decades, even without external events that strengthen militarism.
And how do you get that? Isolationist America, which refused to enter the conflict despite German subs murdering its citizens, is planning to target the German fleet! Which is going to have to get passed the RN before the Americans would be able to get anywhere near it. Or are you suggesting that America is going to do a pearl harbour attack on the HSF? I suspect that you mean the US is going to claim the German fleet as a threat. Given its markedly smaller size, short range, the fact the Germans will probably have to disband a lot of it and again it would still have to get passed the RN that is rather farcical.
And the idea that the USN would be used against against Britain is far more farcical. (though increasingly less so, if the Anglo-Japanese alliance continues) Fact is that the U.S. and Britain have by the 1920s no strategic conflicts, a compatible China policy, and numerous common strategic interests. Despite isolationism, the U.S. is a far more useful potential ally against Germany than Japan, which has endless strategic conflicts with Britain, and is a natural enemy of both Russia and the U.S., both of whom are natural allies of a Britain in conflict with Germany, on top of being far, far away.
Did Germany win. It had to give up all its gains in the west while those in the east could prove very expensive to hold. It's not, if I read the OP correctly, going to get back its extra-European colonies. Also with the war over its going to face the problem of war debts. Both the financial costs of the war and the desire for social change that the population will demand now its over, especially if the military are gaining victory.
Similarly have the liberal democracies lost? Their still standing and fully under the control of their governments with the German withdrawals. Imperial Russia has collapsed and Ottoman Turkey and the Hapsburg empire might well be in their death throbs. To the outside world its going to look like the autocracies that are faltering. [An exception here might be Italy but its probably still not seen as a major power as well as a less than stable one.
All comparisons are with OTL. Clearly Germany did not lose unlike OTL, and thus militarism will be far less discredited (as little as that was), and the boost to democrats that led to the short lived Taisho democracy will not occur. Yes, Russia was an autocracy, but it remains one of the 3 main Allies. Similarly, even with a mostly status quo ante bellum peace in the West, France will be crippled, thus knocking out one of the pillars of liberal democracy in this period. Most importantly, regardless of what the truth may be, a peace where Germany was able to knock out Russia and force France out while retaining A-L will be percieved in Japan as a German victory, which is all that's important here.
Also, I repeat what I said earlier. The threat the Japanese will see is from the US and it will require ships not divisions to seek to counter that.
And I repeat that this is no different from OTL. The only differences from OTL going into the treaty renewal/WNT period, are factors that would weaken the pro-British faction.
All the figures I've seen are for 32 or 32.5kts for the final design. Admittedly we will never know as it wasn't built but its unlikely to be more than a knot out at most. Even so as I say the difference is a hell of a lot of propulsion, as well as armour and related matters. [Since a longer hull needs more protection].
Well, the best comparison is with contemporary designs. Compare the Lexington battlecruisers, for instance, which have 20000 extra shp, a longer as well as narrower hull, and ~5000 tons less displacement, leading to a projected max speed of around 34 knots. However British estimates were that the design would not make more than 32 knots in practice, which is born out by the speed of the later carrier conversions which had the same hull and shp, but ~10kt less displacement.
As for just how much you can save by cutting down on the propulsion, the average cost of the Rs was only 2% lower than the average cost of the QEs, and the Rs had only 53% the shp. Or you can look at the 7.5 million pounds for the Nelson, which had a lot more cut than just the reduction to 28% the shp.
Japanese for whatever reason, not completing any of their 18" ships.
Well, I think that's a pretty safe assumption, considering the 1923 Earthquake. Certainly, they wouldn't be laid down for some time.
Actually it makes a lot of sense. Possibly not so much in terms of a full sized fleet battle as that would probably have been seen as highly unlikely. More likely in a ship or small squadron operating on their own as so often happened in WWII. [Although could be using hindsight here]. The extra speed means that more enemy ships could be caught or, if outnumbered, escaped. Even with a full fleet battle having powerful units who could choose the range and accept and decline combat from a single slow block like the US standards could be very useful. Or being able to cross the enemy line as the Japanese were able to at Tsushima. Also the higher the cruising speed the better a chance of avoiding air or sub attack, the later already having been shown as significant.
Well, like you said, the benefits of superior speed for individual action is with hindsight, while limited benefits of a nonuniform battleline are readily apparent. The N3 design was slated for 23 knots as well, and I see no reason for Britain to go for 26 knots in a nonescalatory N3 replacement. Well, that and the impossibility of building a Super Lion analogue for cheaper than the G3. Plus 26 knots is not nearly enough of an advantage for the kind of tactical advantages the Japanese got at Tsushima, especially not with the (well known) tight tactical radiuses of U.S. capital ships and the long ranges of post WWI capital ship combat. Even 30 knots is probably insufficient.
Quite possibly. I didn't say the price was accurate but just that severe deflation did occur in the 30's. Also don't forget that with Vanguard the main armament wasn't totally free as the guns had to be reconditioned and as you said there was probably a sizeable measure of wartime inflation. As such while the figures you quote may well be too low it may not be too far off.
Actually the number I cited as exclusive of main guns, mountings, and turret armor. I figure the actual cost of the Vanguard is probably in excess of 10 million, even with reconditioned guns and mountings much cheaper than new. There are also other examples like the 11 million pounds for the reconstruction of the QE during the same time period the KGVs were being built.
As I've said elsewhere the British yards were very, very cost effective for much of this period, especially compared to the US.

And as I 've said elsewhere, the cost of the later Standards strongly suggest otherwise, if not point in the other direction altogether.