A different Cold War

Very much so. It is far from implausible to imagine, and I like to picture, a Cold War reunion of WWII Allied and Axis veterans (e.g. the 20-year anniversary of the end of the war) that includes Patton (surviving because of butterflies), Monty, Rommel, Messe, and the stand-in for De Gaulle sharing drinks and memories, and wistfully regretting political circumstances prevented the Allies and the post-Nazi Axis (and themselves as their generals) from joining hands at the end of WWII, liberating Eastern Europe, and putting an end to the Communist beast then and there, sparing the world a lot of trouble. Of course, once the Eight-Week War happens, the general mood is going to be the West at last set right this wrong for the Eastern Europeans, although it is a pity the threat of Soviet nukes again prevented them from riding to Moscow and finishing the job.

Almost surely Brussels for the same reasons as OTL, only more so. As a matter of fact, it is quite possible and perhaps even fairly likely ITTL Belgium eventually gets its internal issues settled by peaceful partition between the Netherlands and France, with Brussels becoming the EU's federal district capital, just like Washington D.C., Brazilia, and New Delhi. Admittedly I have no really good idea about the capital of the EAU. I welcome suggestions, although in (geo)political terms I think a federal district carved out someplace in Indochina, Malaysia, or Indonesia would be fairly fitting and plausible. On second thoughts, perhaps this would be an appropriate outcome for TTL Singapore.

Not an independent state ITTL, although by becoming the federal district capital of the EAU, its global importance would be even bigger. It would become one of the world's major centers. Washington and Brussels would mostly be political capitals, since the role of major economic hub is filled in America and Europe by several other big cities. But Singapore has the tradition and position to be a trade hub, too, so perhaps it would do a little of both. As a matter of fact, the USA, EU, EAU, and India are so huge, popolous, and wealthy that in many regards they are going to be fairly polycentric. There would not be any of the massive functional centralization of say Paris for France.

Good question. Let's see:

The USA has a significantly bigger population than OTL, because of its annexation of Canada and Australasia (quite possibly part of the Caribbean and Central America, too) and getting substantially more Chinese immigration on top of what it got IOTL from Asia and Latin America. The EU has a significantly bigger population too, because its TTL gains (West Balkans, Switzerland, Norway, Ukraine, a lot more Chinese immigration, probably a little more immigration from Russia) compensate its losses (Britain, Ireland, no large Muslim immigrant community) and then some. Thanks to their prosperity, both powers can absorb a significantly bigger amount of immigrants (provided they can be integrated w/o excessive difficulties) w/o too much socio-political backlash. The more progressive attitude of the USA likely means, among other things, significantly more Hispanic immigrants get legalized, although their number is not necessarily higher.

The EAU probably has a little more population than the OTL sum of its member states, thanks to a more prosperous northern Korea and Indochina that don't suffer the damage of Communist rule and Cold War conflicts (OTOH the end of WWII is a little more damaging for Japan and Indochina suffers large-scale anti-Communist purges similar to the ones of Indonesia, but these negative  factors are likely of lesser impact than the positive ones), as well as the Chinese refugees that didn't find their way to the Western powers. OTOH, this area hasn't been very friendly to economic immigration, so they probably didn't take much more than what the Chinese disaster forced them to, and most of their other immigration is internal. Southeast Asia begins its industrialization a little earlier, so they start their demographic transition a little earlier, too.

India is bigger thanks to no partition and probably has more or less the same population as OTL China or its OTL self plus Pakistan, Bangladesh, and Sri Lanka, due to earlier and greater industrialization slowing demographic growth, and population control policies that cannot be so effective as the heavy-handed one-child policy of OTL China in a democratic state, and in all likelihood aren't much different from OTL. For various reasons, they probably don't get near as many Chinese refugees as Southeast Asia, and later the USA and the EU. TTL India does a much better job than OTL Pakistan in managing the Muslim-majority areas of the Indian subcontinent, probably including population control and greater economic growth. Indian Ceylon is spared the damage of the civil war.

China is the big loser in this scenario, since they lose a lot more people than OTL to wars, WMD attacks, famine, and population displacement to the USA, EU, and EAU. On the other hand, the nationalist government probably sees no big reason to pursue population control policies, and might even do the opposite, either to compensate for the losses or for perceived power reasons (like OTL Iran). I'm doubtful this would be enough to entirely make up for the losses, though. They lost a lot more territories than OTL, but they either were low-populated or its surviving Han population was forcibly transferred to China proper, so it does not make much of a difference.

Russia is another serious loser ITTL, although not so bad as China, from harsher Stalinist mass purges and genocides, a more screwed-up economy, and a messier, more violent collapse of the Soviet regime. The Russians may well try natalist policies at some point, although their effectiveness is questionable. There is in all likelihood more Russian immigration to Europe and America.

Africa and the Muslim world probably aren't much different from OTL. South Africa has a bigger population thanks to its annexation of the rest of British Southern Africa and its greater prosperity. The Maghreb may or may not be substantially different depending on circumstances (i.e. they get the Islamist bug and fall to a state not much different from OTL, or they avoid it and get significantly better thanks to ongoing close economic ties with Europe). Latin America probably isn't much different from OTL, although they are little better off thanks to greater trade with the USA, EU, and EAU, a little less civil wars and screwed-up regimes.

Last but not least, although this is yet in the near future of the TL: in a few decades at most, large-scale immigration from Earth to the space colonies is going to start. The availability of nuclear-pulse propulsion and fast-breeder reactors as mature technologies, and more so commercial fusion power in a decade or so, not to mention various kinds of progress in bio-, nano-, and materials technologies, means to move a steady flux of millions or dozens of millions people to space is economically and technologically feasible over the time immigration usually takes. It is just a matter of building the necessary colonies and transport infrastructure, and a little more down the line the terraforming equipment. This is precisely what the world powers are gearing up to do in the early 21st century.

It did; China broke up after the Sino-Soviet War and even after the bandits, minor warlords, and ethnic separatists got wiped out after a while, it stayed divided between a PRC northern rump, a southwestern Taoist Taiping equivalent, and a central-southeastern RoC (which used Taiwan and Hainan as the main power base) for most of the Cold War period. It's just like all the other periods of division in Chinese history, it was temporary and ended by the 21st century because Chinese national identity was just too strong once the nation really started to recover from the disaster and the KMT re-unification bid was eventually successful because they were a rather better alternative than their rivals (the CCP was thoroughly discredited because of Stalin's and Mao's actions and the disaster they brought to the Chinese people and after the war the surviving leadership was made up of Communist hardliners, the Yellow Banners were theocratic, reactionary thugs) and the RoC had much greater resources thanks to its intact insular powerbase and Western support. Unfortunately, while the Nationalist leadership stayed relatively moderate as long as it was made up of the KMT old guard that had taken refuge on the islands, over time and with reunification power inevitably shifted in the hands of mainlander younger generations that had been traumatized by these events into nasty, paranoid hyper-nationalism. Much like what happened to post-Soviet Russia if to an even higher degree. The dozens millions Chinese refugees that found their way to immigrate in the Western powers experienced much better living conditions, as usual they were relatively easy for the host nations to integrate (and general prosperity in the West made it socio-economically tolerable), so they developed an entirely different, much nicer and more positive attitude than the ones that stayed back.

China also suffered major territorial losses (Tibet, Xinjiang, Manchuria, Mongolia) that are in all likelihood permanent in the foreseeable future given the ethnic situation, short of them winning a major war against India or Russia. This is extremely unlikely with India but perhaps doable with Russia, which is also screwed-up but not so much as China, but has a much lower manpower. The world powers may or may not tolerate China and Russia keeping or developing a WMD deterrent, but it would be far from implausible for them to act and prevent post-Cold War China or Russia from having it, given the two powers' instablity, rogue attitudes, and the scary precedent of the Sino-Soviet War. OTOH, China automatically got reunification with Taiwan and Hainan.

It might go either way (reunification with the RoC or foreign rule) depending on political butterflies and exact time schedule of diplomatic dealings, but independence with expansion on the mainland isn't much feasible (it would sooner or later end in annexation by the RoC which is doing the same thing with rather greater resources). Tentatively I would assume the most likely alternative is the British were willing to return it to the RoC once end of the lease got near but the Hong Kong inhabitants were hostile to reunification with a screwed-up China and London would not coerce them on this. Later, when Britain itself became unable to sustain its remaining colonies because of its own isolationism-wrought economic decline, the Hong Kong people asked to join the EAU, which was willing to take them since they already had a sizable Chinese minority in Southeast Asia (a mix of long-existing diaspora and former Cold War refugees). The British did not object to this solution, the Chinese vehemently did, but they were not strong enough to oppose it by force against superior EAU power. Hong Kong became another source of tension between China and the EAU besides the South China Sea islands.

Macau is an analogue/extension of HK in many ways, most definitely including the unwillingness of its inhabitants to be returned to Chinese rule, except unlike declining Britain the EU superpower certainly has more than enough the economic and military resources to sustain its rule in the exclave indefinitely if they so choose. The main point is, are they willing to do so ? They have been so exceedingly successful on the global (and interplanetary) stage in so many ways that they don't really need the tiny, faraway remnants of their member states' colonial empires to feel important. The Pacific is kinda outside their main sphere of interest and they have kinda subcontracted management of the region to their EAU and to a lesser degree US allies. So it is quite possible they decide to encourage cession of Macau to the EAU just like HK, a solution the locals would certainly accept. Depending on political butterflies, it might go both ways: either Macau stays under European rule indefinitely or it is ceded to the EAU and gets merged with HK. Perhaps if the EU can reap some significant trade benefit from continuing ownership the former outcome gets more likely, otherwise the latter is more probable. Hyper-nationalist China definitely won't like either solution and shall make its displeasure loudly known, but they can do little more than empty threats, complaints, and perhaps the small, petty acts of harassment OTL North Korea indulges to.

One thing the rogue big regional powers (Russia and China) cannot really afford to do ITTL is the amount of bullying the Western powers their OTL counterparts get away with, given the much greater power differential. If they were to threaten GSO strategic interests or escalate tensions too much, the US-EU-EAU team-up could easily utterly crush them on the battlefield in a conventional conflict and occupy them in a few weeks to months. As we said, it is even rather questionable the Western world powers even let the rogue powers keep or develop a WMD deterrent (as a matter of fact, such a contingency might well be a good casus belli). Of course, occupation of China or to a lesser degree Russia (which is just as big, but much less populated) would be a most serious (but ultimately manageable if need be) effort for the GSO, which is way they don't escalate tensions on their own for trivial reasons if they can avoid it, especially since they have to deal with a third rogue area of the world, the Muslim word. But you can definitely expect at least something like extensive bombing campaigns whenever the rogues overstep their bounds and seem in need of a smacking. Then again, global terrorism pulling something like 9-11 or a rogue state's attempt to develop WMDs would certainly motivate the GSO at full-scale military intervention and occupation, even more so if the culprit is something like Iran, Turkey, the UAK/UAR, or Egypt instead of Russia or China. Something like Afghanistan goes without saying, it would be a relatively trivial effort for the GSO. Depending on the situation, you might even get India interested and involved in the peacekeeping effort: e.g. it would be plausible if the problem is China or the Muslim world.