I wonder if @JLowder can comment on this, at least the gods thing if not the "doors" or "bridge" part since he was there.
There isn't, as far as I know, a current official cosmology or single explanation for all this, but the company has offered various options through official releases such as
Manual of the Planes and the Planescape setting over the years.
Whether travel between worlds/settings in the game is easy or not depends on your campaign. It's been an active discussion for years. One of the issues raised by designers when Spelljammer was being put together was the potential damage linking the worlds could do to their individuality. It looks to be the same concern you are raising, and it's a legit question. But the solution we settled on back when Spelljammer was first released was to make the journey itself difficult and dangerous, so travel between the Realms and Krynn, for example, could be relatively rare if that's how you wanted to play it. If you as a GM wanted travel to be impossible in your campaign, to keep the worlds distinct, it's easy enough to come up with an in-continuity reason for that. Spelljamming isn't a thing and the way magic works on the different worlds makes traveling by gate quite dangerous and tied to potentially deadly or horrible side effects you don't want to risk. If you want some stable gates or bridges, they can exist, but you can throttle their use as desired through plot and associated game mechanisms.
With the gods, my baseline was always that the mortal characters, even the most powerful PCs or NPCs, could understand the divine only up to a point, after which it required
faith from them to interact with or accept the divine. That means mortal characters with imperfect knowledge, which is a powerful game and narrative tool. (For Ravenloft, the one hard and fast rule we stuck by was the Dark Powers and their motivations were unknowable. They may or may not be divine, but they are distant and their goals a mystery. That's one of the things that helped make the setting suitable for horror stories.)
Inasmuch as
Prince of Lies is a place where the official releases explore that concept, that's foregrounded. It's part of the reason why I have the chapter with a god doing multiple things at the same time, to emphasize the "alien" divine reality or the chapter with Adon, who had been a friend and companion of the characters who were raised up to be gods, standing in the asylum, unable to figure out why the deities don't just end the suffering. He's mortal. He doesn't get how the hierarchy and politics and divine motivations work. In
Ring of Winter, Rayburton gets caught up in celestial bureaucracy after he dies because different gods hold sway over different parts of the Realms. The whole system is not clear to him. Then again, maybe Ubtao in Chult is also related to Ao in the Heartlands, as the names suggest. (There is no official answer there, but I named Ubtao to create the possible resonance. Possible, not defined, though.)
So, sure, it's possible Tiamat could have worshipers and a presence on multiple worlds. But that could also be the reason why travel between those worlds is now hard. The local gods do not want another deity from elsewhere to land potential worshipers and gain a foothold. They saw what happened with Tiamat. You can also build your campaign so that the God of Knowledge is the same across worlds, just using different names to fit with different cultures. Or people think that the god is the same and are wrong, thanks to imperfect knowledge and/or divine deception. One of my favorite things about working on projects like
Prince was exploring that gap and the notion of faith, and I tried to leave the big questions open because it's more interesting that way. (And I've had some great conversations with other Realms creators on the subject. Bob Salvatore and I had a long phone call after
Prince came out where we discussed whether or not there could be atheists or agnostics in a world like the Realms where the gods regularly manifest--I firmly believe the answer is yes because of the imperfect mortal knowledge or the possibility some gods are kind of grifters, as
Prince suggests, and Bob was surprised I took that route. He liked it but hadn't considered it.)
How it all plays out for your campaign depends on what you want. Ideally, Wizards would be smart to give people options and tools to make it possible for them to build the campaign they want without dictating a specific style of play. The company's decision to abandon formal canon is a step toward doing that, I think. So if you want your worlds distinct, build those walls and make the walls and their functioning part of your story, if that works for you.
Hope that helps.
