I don’t believe a deity can be a “greater god” in one world and anything else in another—that is, I believe that when the 5e Dungeon Master’s Guide refers to “greater god,” it refers to what previous editions called an “overdeity.” However, it is difficult to back this up from within 5e—as far as I am aware, nothing goes into more detail on the questions you ask in that edition.
Previous editions go into far more detail; sometimes deities even received actual statblocks (though these were usually later ret-conned as statblocks of various avatars, since of course it would be problematic to really pin down a god’s stats). In particular, more different divine ranks were recognized—in 3e, Deities & Demigods defined divine rank as a number from 0 to 21+, and named tiers of them, for example. Even just the named tiers had more granularity—3e recognizes “quasi-deity,” “demigod,” “lesser deities,” “intermediate deities,” “greater deities,” and then 21+ were described as follows:
These entities are beyond the ken of mortals and care nothing for worshipers. They do not grant spells, do not answer prayers, and do not respond to queries. If they are known at all, it is to a handful of scholars on the Material Plane. They are called overdeities. In some pantheistic systems, the consent of an overdeity is required to become a god.
This matches the 5e description of “greater gods” pretty well, and it’s certainly true that overdeities—such as Ao, who sets the rules in the Forgotten Realms—are not found anywhere in the planes, and do not trouble themselves much with the scheming and meddling of the gods beneath them.
Under the same rules, “greater deities” certainly were physically incarnated within their own divine realms, could theoretically be killed if you went to those realms and defeated them there (but even a considerably-stronger god would have an extremely hard time invading and destroying a god within their own divine realm, and usually a god’s death was the result of a great deal of scheming to weaken them first).
But 5e doesn’t use those terms. There aren’t 21+ divine ranks, there aren’t “intermediate deities” or “overdeities,” there’s just three ranks, quasi, lesser, and greater.
In general, it’s often not clear how much 5e means to ret-con, and how much 5e is just simplifying its explanations in products that are not directly devoted to the nature of some particular piece of the lore. The 5e description more or less matches the previous description, just with less granularity, but it’s not necessarily certain that’s what’s going on.
So too with “greater gods.” The description seems to match that of overdeities, to be sure. The Dungeon Master’s Guide doesn’t name any of these greater gods, though, so it’s hard to be certain. Lolth—mentioned as a lesser god—was a greater deity in the past, but that was in 4e, which makes it difficult to consider much. Prior to that, she was an intermediate deity, a level that 5e doesn’t even recognize.
And then there is the claim that deities have different ranks in different worlds—which is certainly consistent with previous editions—but then only defines three tiers, one of which is, maybe, that of an overdeity who certainly aren’t found at another rank on another world. As you say, that would be a contradiction in terms, since the definition of “greater god” has them without a physical body and lesser ranks have them. So perhaps it meant only that gods could shift between lesser deity status on one world and quasi-deity status on another, but it’s strange to say that.
What makes the most sense is to take that line as referring to greater granularity in divine rank, as was found in previous editions. That “lesser deities” are, in fact, broken down into multiple, say, “sub-ranks,” and it is among these, and demigod, that a god may have different on different worlds.
As to your specific questions, then, under this understanding:
So if a God has a greater deity rank in one world in the multiverse, and lesser in another, what are the implications of this?
Under my understanding of things, this is not actually possible. Ao is only relevant to the Forgotten Realms, where they are the overdeity. In the Forgotten Realms, Ao sets all the rules for gods and mortals alike. Outside the Forgotten Realms, they are a non-entity. In many ways, overdeities aren’t really that much like deities—it’s perhaps more helpful to think of them as something else entirely, “over” deities certainly, but not actually deities themselves.
This makes your first question moot. Onward then
Are these ranks only defined in the actual worlds, in the outer planes these ranks don’t matter/exist?
In past editions, they certainly exist, and matter a great deal, but the rank that a deity has in the Outer Planes depends on their rank across many worlds—the greatest gods found only in one world are minor players in the Outer Planes, quite possibly less than a minor deity found on many worlds.
To be frank, though, I’m not sure any edition of D&D has ever really gotten into the details of how this works. The 2e Planescape setting—which dealt most directly with the multiverse—certainly described which gods were the biggest players in planar politics, but political power in the planes is not necessarily directly correlated with personal divine rank (especially since some non-deities are dramatically more influential than many deities).
Are there multiple different entities of the same god?
Sometimes! Usually not, but it has happened. In particular, if a foreign deity is introduced to a world, it is not unheard of for a native deity to take on the guise and name of the foreign deity as one of their aspects, so that the faith devoted to the foreign deity actually goes to the native one. This is, as I understand it, basically a question of who notices what’s happening and takes initiative first. In a case such as this, the being worshiped under the foreign god’s name is not actually that god, and is a separate being from the foreign god.
Note that this also has happened with gods within one world. In particular, Shar, from the Forgotten Realms, has masqueraded as, and swiped the belief in, several gods; it’s something of a signature move for her. This would be the sort of scheming that weakens a god and makes it possible that another might kill them—another Shar signature move.
In the multiverse is the Lolth worshiped in Greyhawk settings, the same being as the Forgotten Realms worshiped Lolth?
Yes, at least in prior editions, they were very much the same. Unless things have changed dramatically, it’s very unlikely that any deity is going to try to pretend to being Lolth. Even Shar would likely think twice before crossing the Demon Queen.
If Tiamat as a lesser goddess is slain in the Forgotten Realms, in another world where she is worshiped is she still able to hear/grant prayers and spells?
“Tiamat” cannot be slain “in the Forgotten Realms.” Tiamat’s avatar, definitely, but not Tiamat herself. Gods don’t show up on the Material Plane in person, basically ever. In fact, they don’t usually leave their own divine realms much at all, and doing so—except in the cases of a few deities of wandering who have no fixed realm—is a good way to piss off pretty much everyone. This is what avatars are for. (Actually, after the “Avatar Crisis,” even sending an avatar to the Forgotten Realms is pretty frowned upon, including by Ao.)
So if you want to kill the true Tiamat, you have to go to Hell and kill her within her lair. If you do that, then yes, it’s for everyone. Her clerics will no longer be able to prepare spells (though dragons rarely bother with being actual clerics even if they’re devout).
If you want it to stick, though, that’s harder. Upon her death, Tiamat’s divine essence is still going to exist. The belief of her faithful is still going to exist. So long as both exist, Tiamat may well come back, death or no. Someone could claim her divine essence, whether that’s another god swooping in and taking it (see Shar), or a mortal using it for apotheosis (many examples), and that would certainly put a damper on things for Tiamat, but if the belief in Tiamat—not whoever took the essence—still exists, Tiamat may yet still return to reclaim her essence. This happened with the Forgotten Realms god Bane, who was killed and whose essence was claimed by his son Iyachtu Xvim. Bane’s followers begrudgingly worshiped Iyachtu Xvim, but basically all of them preferred Bane and wished they could have him again. That was enough for Bane to reclaim his essence and return from the dead (killing Iyachtu Xvim, but that wasn’t much of a concern for Bane).
The real thing that kills a god is for people to stop believing in their divinity. So long as that persists, a god can always come back. Coming back will take time, and cost them dearly, and it’s a risky proposition—because while they’re gone, their worshipers aren’t getting divine intervention in the form of miracles and divine spells, which will erode their faith, and there will always be another god ready to step up and fill in.
What if she has a greater deity rank on another world?
Overdeities don’t really exist in any fashion that can be killed, but it’s conceivable for one to “die” if their world dies. Planescape, and even more so another 2e setting, Spelljammer, have hinted as worlds that have died. There’s even, if I remember correctly, a broken crystal sphere (without getting into too much detail, a crystal sphere is what an overdeity actually has control over). If that happens, the overdeity in question may well be dead.
Also, for the record, as a big fan of the original Tome of Magic supplement that first described vestiges, defining them as deities, quasi- or otherwise, is incredibly misleading and unhelpful, in my opinion. They are not the same as dead gods, though a dead god might well be a candidate for becoming one (e.g. Amon, Tenebrous, though the first is uncertain and the second is a very special case). But again, is this ret-con or oversimplification? Hard to say.