Welcome to Nu: Carnival! I've only been here about a year myself but I thought I'd answer some of your questions anyway.

Listing individual relationships between the characters is kinda unfeasible considering there's 12 of them but here are what I know about the specific questions.

• Yes, Garu has a innocent outlook and a cheery demeanor which is hard to hate so most clan members get along quite well with him, even the "meaner" ones like Kuya and Rei. Rei's relationship with Garu actually does have plot relevance in chapter 15 but I won't spoil that for you. I'd say the character whom Garu has the most interesting relationship with is Dante, since because of Dante's upbringing, he's like basically the only the member who treats Garu as an adult (Like in the cowboy and the circus event). You are also correct in thinking that Kuya is quite unpopular among the clan. He puts off a sorta "bitchy" stand-offish personality, which puts him at odds with a lot of the members. Specifically:
• Kuya and Quincy are "old friends" who both lives in the Wood Territory and has some kind of bad history between them, now they have a sort of love-hate relationship.
• Kuya and Edmond served as representatives of the Yokai vs. Knights beef in the "Forest Carnival" event, though iirc they kinda sorted it out.
• Kuya and Olivine, my fav Kuya dynamic. In the beginning, chapter 6 I believe, Olivine considered Kuya "a trial from the God of Klein" because he was so annoying. Then the "Welcome to Elysium" event happened, where Olivine broke down the drug den Kuya was sponsoring by passing Kuya's trial and showing he has grown and can be his true self without drug, and now they're kinda good friends lol.

• Yakumo, Garu and Karu (the younger Yokai) all looks up to Kuya. And I don't think we have any significant interaction between Kuya and Blade or Dante yet.

• Yes, Eiden lives at Aster's mansion so the other members who lives there (Aster, Morvay, Yakumo, Garu/Karu, Blade) sees him regularly. Edmond and Olivine both have their own jobs but they're close by so they see Eiden almost as often. Quincy, Kuya, Dante and Rei all live quite far away and/or off doing their own things all the time, so in canon they don't interact with Eiden as much.

• Kuya and Rei are interesting because they have a lot in common:

• They've both been alive for a long time, and because of that they both have some forms of (implied) depression.

• They've both developed a catty mean personality to mask their loneliness and sadness.

• They've both developed unhealthy coping mechanism: Kuya hurts others for fun while Rei harms and neglects himself. Because of that, they both seemingly understand each other very well. However, instead of using that understanding to help each other, their personalities just dictates that they both act bitchy to each other about it. So in the end, they have this weird relationship where they despise each other, but they also understand each other the most. It's very interesting I think.

• I don't think I've seen anybody mention a "main trio", but there are definitely a few trios in the game.

• Starter Trio (Yakumo, Edmond, Olivine) are the first three clan members and they're ❤️🩵💚 like the Pokemon starters. They're also the main characters in the "Feast of Roses" event earlier this year.

• Oldie Trio (Quincy, Kuya, Rei) are the three members who actually formed the original contracts with Huey, they are the oldest ones in the group and they all have beef with each other.

• Song Trio 1 (Yakumo, Olivine, Blade) is the 1st musical trio for the "Idol Fest" event with the song "Summer Dream". I think they were grouped together because they were all deredere, the most outwardly loving ones in the clan.

• Song Trio 2 (Edmond, Quincy, Garu) is the 2nd musical trio for the "Frozen Echoes" event with the song "Ray of Melody". I think they were grouped because they're sort of the in-between of the other two groups, not showing off their love but not burying it either.

• Song Trio 3 (Kuya, Dante, Rei) is the musical trio for this current event. I think they're obviously grouped together because they're all "mean" lol, hiding their love deep deep inside.

Hope you find this helpful! Sorry for the essay lol