I don't think people understand the absolutely driving force 'Sea to shining sea' really was. Britain and France wasnt going to be able to check it if they tried. Manifest destiny. It was a National obsession and woe to the fool who got in the way of it. It was going to happen one way or another. The closest thing we have today to that in the modern day was the Space Race against Russia. We as a Nation were obsessed with it and with winning. Today? I hate to use this example but Trump Derangement Syndrome is the closest thing I can think of but even that is a bit off the mark. Now imagine the entire country swept up in something like that but focused towards a singular goal we are going to achieve no matter what? You begin to understand what our expansion westward means.
I think maybe the immediate aftermath of 9/11 when the whole country was dead set on going after Osama is another example. But the broader point I agree: it was something that was heavily favored across the political spectrum. Trying to limit that was only making things worse, especially when you consider the fact that most of the colonists were basically the misfits of Europe that nobody really wanted.
I generally agree, although part of the problem with George III was that he actively was taking a role in governance. But had the British had the sense to realize they couldn't turn it into Ireland writ large (and they couldn't, since unlike Ireland the Americas existed as a dumping ground for people they didn't want at home), then they'd have been better off taking the approach they did with Canada: Let them generally run things with only limited supervision from London, and otherwise let them govern as free subjects.
So ennobling a bunch of Americans really wasn't going to cut it, nor was stringing them along. Something the British didn't realize until too late, along with the fact that literally all of the major powers in Europe were only too happy to dogpile them for their idiocy.
Had they done so, though, it raises some *really* interesting possibilities. America might have become a dominion in its own right, and if so that massively changes the world order. An America behind Britain means they suddenly have a country able to help throw massive resources behind them in the event of emergencies like the Napoleonic Wars, and World War I might not even occur because even though the Germans had a larger industrial base at home, America is now a heavily industrialized power on the side of the British at the outset. So a combined British-American-Canadian war machine (because it's not like Canada has to worry about a fellow Dominion attacking them) would likely be way too much for the Germans to overcome.
What happens with India, I have no idea; the British had no real way to integrate it into the rest of the Empire since "divide and rule" was at the heart of their strategy for holding it in the first place.
Another interesting question is Ireland...if America is the jewel of the crown instead of India (and given what we achieved OTL I think it's likely America largely develops as it did sans slavery), then it puts us in a position of being able to exert a serious degree of influence on London's policies, especially if we're home to a large Irish diaspora.
The easiest realistic way for the British Empire to still exist in a certain form is for an Imperial Parliament to be formed representing the White Dominions initially, instead of their being granted self-government. This was a very serious proposal with a lot of support. Even if the United States not a part of it, such a plausible Imperial Federation could consist of the UK, Ireland, Canada, Australia, New Zealand. The question is whether or not it can also successfully integrate South Africa. It would also plausibly include smaller colonies which have reason to remain associated: Singapore, Penang, North Borneo, Hong Kong (depending on the circumstances), Mauritius, Guiana, British Honduras, Jamaica, Malta, etc.
There is no question a Federation controlling the territory above would still be an Empire in every sense of the word, and a superpower.
That's a difficult question. They still probably get independence or whatever in 1910, and I can see them being dicks regarding apartheid. The difference is, without a World War II to completely discredit things like racism and eugenics, the rest of the world might not care. And I don't see a World War II really happening in this scenario, at least not one that unfolded the way the original did.
Ultimately though, I think it would largely drift to bring what the Commonwealth is today; without a unifying external threat (like the French, the Germans, or the Soviets), things drift apart. Not violently, but enough that it's rendered ineffective.
Also, assuming they cough it up in the first place, I don't see the Chinese getting Hong Kong back. And if anything, any Communist insurgency likely gets beaten back by the Commonwealth and their friends.
Get the Dutch and the Japanese on board and you have a proto Earth Alliance. *Nobody* is going to want to take that combination of hard and soft power on. Although we end up misspelling everything so that's one downside...