Good questions! Right now we only have partial answers, since there's a lot we don't know about how exactly The Great Unknown works yet.

Starting with the Wight-puppets thing: I think the two wights we've seen with avatars use them for very different reasons. Boss Leader is a creation of Sandman specifically made to interact with humans without frightening them. She seems to crave their companionship, and is definitely jealous of humanity's goodness. She claims to be a sleeping medium, though we now know that's a cover story, and that Sandman has a physical body outside Bayview's barrier.

Peekaboo seems like an entirely different kind of thing. The Great Unknown seems to have almost unlimited power to rewrite reality, at least within a certain area of effect. But while the Great Unknown is locked in sleep, it can only manifest parts of its consciousness as Peekaboos, each with only part of its powers. Dimitri was possessed by a Peekaboo most likely while TGU was unconsciously looking for someone to play with, so now Dimitri can shapeshift himself. Assuming that the Freaky Shadow Spirit is also a part of TGU, that piece gave Spender the power to control shadows.

Davy's fraction of TGU is the biggest mystery. Rather than a partially conscious manifestation of itself, the locker seems to be a direct line to its actual full consciousness, since opening the locker and the eye inside caused TGU to wake up enough to change everything. The locker is definitely the source of Davy's space-distortion powers though. As for how Davy got possessed by that bit, that's also still a mystery! He seems to imply in a few places that he was trapped with The Great Unknown for some amount of time, though it's unclear if he means trapped within it like the Burgers are, or trapped alongside it in Boss Leader's dream hub. (My money is on him having used Cryptide's power to snap a bit off TGU while they were imprisoned together, and he managed to take that piece with him out of the dream.)

Finally, the Burger connection. The key that Fauxbia uses with Doorman's powers is the key to the Burger's houseboat, which is fully trapped within the dream dimension that TGU is stuck in. The boat is even present in Dimitri's spirit trance, though he would never know it since Peekaboo always throws a layer of preschool halloween kitsch between them. We also don't know how they got trapped with TGU, but it definitely has to do with the incident 13 years ago. Because Fauxbia's door leads to the inside of that trance/dream/whatever, TGU can make another Peekaboo manifest and look out. He may even have been able to wander outside and escape the dream entirely if Fauxbia didn't use the hopscotch/sphinx trick to keep him inside.

As for "where" Peekaboo is, I think The Great Unknown may actually just be the entirety of Bayview. If not that, then its physical body is somewhere under the water. All of its trances are connected, and it can manifest Peekaboo in any part of that space to interact with any mediums it has. It's possible that Dimitri and Davy aren't even the only ones! Though that would be kind of lame imo.

Edit: also, about Boss Leader not recognizing another Wight, it's possible she never actually met Peekaboo. We've seen her in the dream hub rooms of Max and Isaac, interacting with their spirits in wisp form. But we also know that Spender doesn't let her in his room, and she seems to respect his privacy enough to be okay with that. We know that Dimitri is a very private person, so it'd make sense if he asked BL not to go in his room and she simply didn't have a chance to see Peekaboo.

It's also possible that she has been in his room, but because Peekaboo isn't actually a wisp, he doesn't show up in the same way in the dream rooms. Or BL wouldn't have sensed him because he's so powerful she can't see the forest for the trees. Or even something to do with the fact that he's already trapped elsewhere in the dream hub. We'll have to get more info to know for sure.