Not entirely true. While Planescape certainly followed that model, not all settings do. Forgotten Realms is one of those settings that follow a different model. Dragonlance is another.
Wrong. Sorry to start out on such a confrontational tone, but this is something I feel strongly about (plus, I haven't had my breakfast yet

).
No one knows what is up with Eberron, and I'm fairly sure that the outer planes are out of reach in Dark Sun.
Some times, people like treating the expanded D&D universe as if they were all existing side by side. You can hop from FR to Greyhawk to Nentir Vale to Eberron without problem. Planescape and Spelljammer encourage this. But the simple fact of the matter is that there are far too many conflicting details to ever really work together without handwaving huge portions of those settings away. So many people treat Planescape!D&D as if it was the absolute truth of every setting, but its not. Every setting needs to be looked at through its own lens and cosmology. There is no "subset" when we're dealing with a very specific game setting with its own specific cosmology. There's no shoehorning all the realms to fit one big one. There's a reason we have multiple cosmology options in the DMG - they don't all hold true.
In Forgotten Realms, souls go to the Fuege (spelling?) plane, where they await either pick up by their gods, or judgement by Kelemvor. No "go directly to plane-of-alignment, do not pass Go, do not collect 200g." We are talking about FR only, so it doesn't matter what other settings do. I don't care if you're a LG character in the FR setting - if you're a paladin of Sune (and, yes, they existed in 3e D&D), then you go to the CG aligned Brightwater plane (Sune's realm) upon death, even if you'd go to Mount Celestia in a Planescape game.
No, that very much IS the case in FR.
In 1e/2e it was official as official can be canon that everything existed in the same multiverse. You could go from one world to another in a variety of ways. The deities of Forgotten Realms, Greyhawk, and Dragonlance were all listed as to which Outer Planes they lived on in the same (Great Wheel) cosmology. This wasn't just the rule
if you were playing "a Planescape game"--this was the D&D facts of the multiverse. Playing a Planescape game just meant you set your campaign on the planes and focused on certain themes. The exact same locales and planar connections were D&D canon regardless of what characters or planes your characters (or your entire campaign) happened to be on.
3e took a different angle. In 3e (and 3e
only) each of the campaign settings existed in an entirely different universe/reality. You couldn't travel from one to another (well, there was the obscure possibility of traveling from one reality to another through the Plane of Shadow, but that didn't change that they were fundamentally different multiverses).
5e appears to be taking a middle ground. There is a single multiverse in which all the worlds exist, similar to how 1e/2e did it, though there are a few tweaks to the planes. But inhabitants of different worlds see it differently. Some see the Outer Planes arranged in a Great Wheel, others think they are unconnected domains floating in an Astral Sea, and no one can prove who is right (if any of them are).
That's how it is. That's D&D canon. 3e is the oddball. One can choose to use 3e's take, but that isn't the official way it is now, nor is it how it was for most of D&D's lifetime.
Exactly how to represent different worlds seeing the same multiverse in different ways, from a practical standpoint, is left up the DM. One might say, for instance, that the Faerunian plane of Brightwater, is seen in the Great Wheel perspective as a planar domain that sits on the border between Arborea and Ysgard, with half of it in each, but with the petitioners free to move around the whole place. (I think that might be how 2e described it, but I'm not sure.) A unique NG Outer Plane from the perspective of one world might be seen as a domain somewhere in Elysium, Bytopia, or the Beastlands. And some world simply have no knowledge of or planar connections to certain planes. If it doesn't fit into their understanding of the multiverse, they probably don't even know it exists. Now, if you go out to Sigil you're going to be able to find just about any of these planes, and most natives see them as fitting into the Great Wheel, because it is convenient shared conception that works fairly well, but that is hardly the universal viewpoint.