WWII ends with Allied-Soviet fight

Yep, the Hashemites and the Alaouties are vanilla royal families, same as OTL. The Saudis are a tribal clan. Depending on the circumstances of their ousting, they might retain some significant influence in the Nejd region. However, I do expect the lion's share of power in the UAK goes to Syrian and Iraqi elites, and Arabia is kind of a backwater region within the Kingdom.

Yes, Hussein seemed capable, which is why I assume TTL history's judgement on him/his family would more or less be "he/they honestly tried, did some good, the challenges he/they faced were simply too difficult", since his House is going to lose the throne to Baathist/Islamist revolutionaries twice. Then again, if global NATO eventually restores them to the throne once they are done crushing the Islamists, and it seems like it might stick third time, this might influence historical judgement in a positive way. But this would happen towards the TL's end in early 21st century, when mankind's attention shall be focused on bigger issues, such as the world mostly getting de facto politically unified under G-8/global-NATO, fusion power becoming available, and space colonization getting seriously underway. By then, the switch away from fossil fuels would have turned the Middle East into a marginal backwater of limited global importance once the Islamist threat is crushed.

Most definitely. It is one of the main effects of WWIII. Picture a NATO (working name is Global Security Organization, but I welcome suggestions) with global reach and a self-imposed mandate to be world's policeman, even more integrated than OTL, getting a few UN-style aspects with adjustments (such as voting rights being established by a mix of population, GDP (PPP), and military budget) to make the great powers run the show and be willing to give it real power. Originally it is going to include and be led by a directory of the USA, the EU, the Commonwealth (a seat going obsolete once America and Europe absorb Britain and the Dominions), India, and the Far East Federation. Rehabilitated Russia joins it and gets a seat a decade or so after the war, China does so as well by the turn of the millennium after the theocratic regime falls and it joins the Western world.

Over time the GSO system develops more and more OTL EU-style integration aspects (e.g. free trade, free movement of people), especially between its most developed members, as intra-Western cooperation deepens, the EU federalizes, and the USA and the EU expand to absorb all of their respective continents. Russia stands apart as a close partner of the other Western powers within the GSO, since it is too big and popolous to join the EU as a member state. South America (the parts the USA doesn't absorb) integrates on the model of the USA/EU/FEF and joins the system as another main member. Southeast Asia either forms yet another example of continental/regional integration, or different parts of it are absorbed by India, the FEF, or the USA.

By the 21st century, as most of the world got developed-world or newly-industrialized living standards, well-managed globalization was an outstanding success, and the GSO successfully dealt with the Islamist and theocratic-China threats, not to mention stuff like climate change and a switch away from fossil fuels, the system naturally evolves into the working framework of a de facto world government, as it includes the vast majority of the world's population, wealth, and military power. Only Africa and the Muslim world may still have some serious representation issues about it due to their greater instability, political fragmentation, and backwardness, but they too are eventually going to catch up sometime in the 21st century.

By the time the TL ends, there might well be even a growing reform movement to streamline and democratize the GSO as a world government and make it official, by popular election of its representatives, electing a world president, and so on. In perspective, as Earth gets more and more unified, mass space colonization is underway, and mankind becomes an interplanetary civilization, the next big political issue becomes the status of the Solar System colonies, as well as other typical sci-fi concerns. If one looks far enough, the far future probably looks a lot like Star Trek, only with much more transhumanism (ITTL Nazism fell early enough not to create a cultural taboo against it) and the aliens being an open question (but scientific evidence is making me more and more optimistic about their existence, with Earthlike planets being in all likelihood quite plentiful across the galaxy). ITTL the ST franchise is going to look less like a wishful utopia, and more like an ambitious platform for the next few centuries.