What is Raph in this iteration relationship with his brothers like? Who’s he closer to and does he have beef with any of em?
Raph is... complicated, i suppose. He's got an intense protective sense towards his brothers, but he also has a certain desperation to him?
again, this is a separated AU, and as a protector, i couldn't leave Raphael alone his whole childhood. Hamato Yoshi managed to safe Raph from The Incident and hide him somewhere safe before he tried to go back and find the others. in the process, Yoshi got mutated. as a child, Raph was very very close to his dad and very dependent on his socialization. him and Yoshi actually managed to find a young orphaned human wandering the sewers. she helped them for a while, in exchange for dry shelter and a share of food and amenities. at some point she just became part of the family, and is Raph's sister, best friend, and closest confidant.
Raph grew up hearing stories from his dad about his siblings and mother, despite Yoshi's deteriorating mental state, he still told Raph stories. Yoshi was mutated with a different, later version of the mutagen that is not as stable as the turtle's own mutagen. he later starts having what Raph calls "episodes" where he loses himself to the mutagen. when he's in one of his episodes, he only responds to the name Splinter. Raph has mentally separated his kind and caring Father, Yoshi, and the feral monster Splinter as a coping mechanism.
he eventually discovers his brothers by accident. him and Mikey end up being closest. Mikey, as the Oldest and raised as a caregiver becomes Raph's little safe space where he no longer has to bear the burden of everything going on with his family. Raph, when he meets Leo, is the one that manages to befriend Leo (Mikey and Leo have major beef) and gets him to come sneak out with them quite often, even if Leo is content not to come home with them. Raph can't really give any reason for him to do otherwise, not with Splinters deal, the foot clan closing in, and the state of the sewers compared to Leos cushy life.
as for beef, Raph and Kojiro (Donnie) have major beef. i know that sounds weird, but they're on opposite sides of the Foot - Hamato battle. Raph originally insults Kojiro by treating him like a kid (he was like 12 so he's not wrong but lil guy still feels disrespected) and grows angry when he realized Kojiro is the main reason their first battle with the foot didn't go well, and why Leo cut contact with him and Mikey. its mostly a result of the environments they grew up in. Raph does agree to taking Kojiro and Karai in when Shredder supposedly dies, and learns to care for them, but they don't get nearly as close as Raph is with the others. he does not have the same beef with Karai because she actually remembers her brothers from before The Incident and has never directly attacked any of them.