WARNING: Spoilers for all seven volumes of the manga follow!
Kagami/Gold:

It was said that the only person who can get to Seeu is Kagami. It's true that Kagami seems to be able to talk to Seeu casually and without fear. Kagami seems to understand Seeu, even if Seeu doesn't say a word. Kagami speaks well of Seeu to Kaguya, and often shows concern for Seeu's well-being. Kagami has also specifically said that he died so that Seeu could live.
Seeu cared very much for Kagami. It is hinted that Kagami's death is what drove him to complete madness. He seemed very comfortable around Kagami in the flashbacks we've seen. Seeu himself says that he knows Kagami's face better than his own, and that he thinks of Kagami always, even in his sleep. He's made hundreds of Kagami dolls and keeps their scattered parts in big piles in his palace (kind of creepy!). He also compares Kaguya to Kagami, saying that they were both "equally annoying", though he seems to think this fondly. And also, Seeu constantly dreams of Kagami.
Gold views Seeu as his master and obeys all his orders, even if he sometimes has to go to great lengths to comply to both Seeu and Kaguya's conflicting demands. Seeu obviously trusts Gold very much to put him in charge of Kaguya, and to let him use the Organic Gold.
I'm not really sure what sort of relationship Seeu and Kagami had, since it took place in the past. It's possible that they were extremely close friends. It's also very possible that they were more. It sort of depends on your views on yaoi. If you support it, then you'll probably want to think they were lovers. If you don't, you'll probably want to think they were just friends. Personally, I think they were friends above all else, but maybe there were some sexual feelings between them that may or may not have been acted upon. Regardless, the two of them were extremely close and had much respect for each other.
Kaguya:
She thinks of Seeu almost all the time. When he first appears in her room, she immediately notes his looks and seems to like them (though she's a bit distracted by the huge weapon in his hand ^^;). She quite possibly would have gone with him, had her foster brother and Idou not interfered. She blames Gold's over-directness and lack of subtlety on Seeu, though she seems to think of him fondly. She often see's Seeu in her dreams and once dreams that she chose his world to survive and allowed all the others (even her own) to be destroyed. In another dream, she pleads for him to help her in the difficult times that lie ahead. Also, after Kagami tells her that she can trust Seeu, she makes it her mission to find him. After Seeu takes her to his palace for safe-keeping, she demands that he not leave her completely alone. Her demands include: him helping her clean up the palace, him burying all the non-functioning Kagami dolls, and him staying by her bed every night until she falls asleep. At one point, when he starts to touch her head, she blushes and even asks herself if she was "waiting for something". In a later scene, Kaguya goes back to Seeu's palace and gets oddly tickled when Seeu "tells her off".
Kaguya was the first person to touch Seeu since Kagami. When Seeu is waking up from a dream of holding hands with Take and Serenanede, Kaguya takes his hand. When he wakes up, and realizes they're touching, it freaks Kaguya out. She eventually gets used to touching him, however, as she does so often from that point on. She grabs his arm several times, hugs him, strokes his hair, and lets him lean over and fall asleep in her lap, among other things. She also becomes a "mother" figure for him, scolding him when he doesn't eat his dinner, etc.
Seeu has a strong desire to protect Kaguya, no matter what happens. It almost seems as if he relates to her in some way. While on the phone with her, he asks her if she's alone. It seemed like a strange question, but later events made it more significant. It's revealed that Kaguya feels as if she's completely alone, even when other people are around her, which could remind Seeu of the many years he spent in isolation in his palace. Once he takes her to his palace, he makes sure she is fed and agrees to her demands after a few days. He pretty much does whatever she asks (though she's very reasonable) without complaint. When he almost touches her, his hand trembles. And yes, the fact still remains that he allowed Kaguya to touch him, when even Take-san said that "No one will ever be able to touch him again*. And it wasn't just a one-time thing. Kaguya touches him all the time now, and at the very most, he just seems mildly annoyed. The fact that he can fall asleep in her lap shows that he trusts her.
In the end, Kaguya and Seeu apparently get married. Seeu mellows out quite a bit toward the end of the story, talking more and expressing his feelings a little better, which the other characters seem to think is a direct result of him being around Kaguya. When Kaguya gets the chance to go home to her family and leave this whole mess behind, she chooses to stay with Seeu. The story flashes to a few years later and Seeu and Kaguya are apparently married. They even have a son, who looks almost identical to Seeu (except he has Kaguya's blue eyes). The Asu bloodline can finally carry on.
Waseda Take:
Waseda was originally from the third world and was somehow transported to Asu, where he met Seeu (who was still fairly young and living behind the veil). Though he was told not to go near the palace, Waseda continually went to the veil to talk to Seeu and the two became friends (in a way). Waseda says that he thought Seeu was beautiful, and he argues with the people of Asu for keeping the Prince isolated behind the veil (though this was before he knew about the bacteria). He helps Seeu in his attempts to transfer a human soul into a doll, and after everyone dies off, he encourages Seeu to live on and keep trying. He tries his best to help Seeu in any way he can. After his original body dies and he somehow transfers himself into the body of a doll (that looks like a giant chicken), he spends over 300 years without seeing Seeu. Even after all that time, he tells Kaguya that he'd rather see Seeu again than even his own parents.
Seeu seemed to connect with Waseda more than anyone else on Asu. He immediately entrusted Waseda to being his assistant in the research and experiments, even though Waseda was an other-worlder who could barely speak the language. He opens up (just a little) to Waseda about his concerns of being the end of the Asu bloodline. Later, when the people of Asu are trying to tear down the veil to get inside the palace, Seuu specifically says to Waseda that he wouldn't mind breathing the same air as him. When Kaguya tells Seeu that Waseda is still alive, and is in the body of a doll, it is partly responsible for Seeu's body shutting off (it does that when he experiences very strong emotions that he can't handle). Also, Seeu dreams of holding hands with Take.
In the final volume, Seeu and Take are finally reunited. Sadly, Seeu doesn't remember Take very clearly, though he does seem to have some vague memories of him. Take says that this is alright and that perhaps it is best of Seeu doesn't remember the hardships on Asu all that clearly. Take apparently went to live with Seeu and Kaguya until the doll his soul was living in shut down. This means that Take is more or less dead. I'm sure he was happy to spend his final days (years? The manga doesn't specify how long he lived with Seeu and Kaguya before he died) with the prince he regarded so highly. He was genuinely happy that Seeu had found peace with Kaguya, and says that the hopes of the people of Asu had finally come to pass.
These two had a very interesting relationship. I think this is a wonderful case of friendship. Waseda cared about Seuu very much, and came to love him as he would love his own family. For his part, I think Seuu cared for Waseda as much as he could (given his anti-human condition). I can't imagine anything yaoi going on between them (I know it didn't happen, but I'm talking about future possibilities here, even considering that Waseda could be transferred to a human-like doll). I don't their relationship had anything to do with sexual attraction or romantic notions. It was simply a case of caring about someone very much and admiring them also.
*UPDATE* I'm starting to warm up to this pairing. Though I can't say I support it in the manga, I do enjoy fanworks featuring it. So please don't misunderstand by my above statements. I am not against this pairing, I just don't think it's a "real" pairing in the manga. ^^;
Serenanede:
She was one of the Asu people who Waseda met when he first came to that world. She seemed to be a loyal and faithful servant of Prince Seeu and went to great lengths to ensure his protection from the bacteria. Once she even killed a stray cat that was clawing at the veil, simply because she was terrified of the veil being torn. She volunteers to be the test subject for Seeu's experiments with transferring human souls to dolls. She tells Seeu that, after the experiment, if the doll smiles for him, then that's how he would know it was her. She also said that she would hug him, even though she knew it would be improper. Later, Serenanede tells Waseda that her father had been the head gardener who tended the gardens surrounding the prince's palace, and she says that her wish was to make it beautiful for Seeu as her father had done in the past (though at that point she was unable to, since the disease had caused her hands to fall off).
Seeu cared very much for all of the people of Asu, which is why he worked so hard to find a way to save them. Although he didn't show it, he probably loved them dearly. Serenanede spent more time around him than most of the others, and he seemed comfortable in her company. Seeu dreams of holding hands with Serenanede, which means he MUST have some vague recollection of her, even if it is subconcious. Also, Kaguya (who he is obviously becoming fond of) reminds him of Serenande.
I think this relationship was very sweet. Serenanede loved Seeu, but not in the romantic way. She loved him because she knew he cared for his people. She was perhaps the strongest supporter of keeping the veil up to protect Seeu, and he was probably aware of this. I'm sure that Seeu is most likely to remember her, of all the people of Asu.
Lagunahan:
Obviously, Lagunahan trusted Seeu in order to leave his prized Collapser in Seeu's care. Also, their worlds were allies, so these two men must have been something resembling friends. Lagunahan just seemed like a friendly, kind-hearted person. But he also seemed like the kind of person who wouldn't pry into someone else's life, or try to force himself onto someone. Seeu probably appreciated this. Though we only see them interacting a couple of times throughout the manga (and only in flashbacks at that), I think their relationship is interesting, to say the least. I just wish more information would've been given on it.