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Here's the gist of it; all three of them are half-demon half-human but because Luo Binghe and Hua Cheng were born in the demon realm their demon blood was more dominant, which allowed them to use their resentment/demon energy more easily and with less (or no) issues. However, when their mother was close to giving birth to Wei Wuxian/Ying(?... I noticed that people alternate) political unrest was rising and assassins were popping up. So, she had to give birth to him in the human realm and entrust him to Changze and Cangse ( these two ) that're relatives or friends/acquaintances of the family to keep him safe/hidden. As a result, his human blood was more dominant causing the resentment energy to hurt him while trying to awaken/draw out his demon blood to make it more dominant when he started using demonic cultivation. Basically, his body was tearing itself trying to restructure on a molecule level that went soul deep. After he died the first time, Hua Cheng found his soul to bring him home but it was damaged and in so much pain that he had to make Wuxian fully unconscious so that he could recover properly in a safe place over a long period of time and without suffering. Of course, Wei Wuxian gets brought back to life and when Luo Binghe and Hua Cheng find out they're just like: "What the fuck?! We just want to see our baby brother!" triggered. I'm gonna wrap it up with their family reunion is post canon stuff, Wei Wuxian has his original appearance/body again, and his demon blood doesn't hurt him anymore since it's now fully awakened and he's on par with his brothers power-wise for three reasons 1; he died a second time freeing his soul from it's unintended body, 2; his soul is fully healed, and 3; his family helping him get used to the demon realm and improving his powers via training. Shenanigans follow suit with a lot of fluff and minor sadness.... very minor, 'cause the parents are sick of their kids being in pain lol
Luo Binghe - His job as Demon King is to naturally over see the kingdom/his territory and all the political stuff that entails, not to mention dealing with all the day-to-day conflicts that may or may not occur.
Hua Cheng - His job as Ghost King is to make sure that souls go where they are supposed to, he's basically Luo Binghe's right hand but due to him being equal in power (if I'm wrong please correct me, I'm new to these fandoms plus AUs... so, yeah) and status he earned the title Ghost King. Also, there's no way just one person can oversee/keep track of so many subjects.
Wei Wuxian - His job was/is to be the one to gather intelligence and keep tabs various individuals, locations, businesses, and more. He's also, and obviously, the easy to get along with social butterfly so it ties in to his job but he also serves as a buffer in political talks so like if Luo or Hua is either too blunt, too honest, or something Wei smooths things over and/or swiftly changes the subject and distracts the "offended" party with a random fact and/or compliment to steer things back in to their favor/benefit.
Bonus info: Luo Binghe and Hua Cheng being sent to the human realm is part of their education such as: learning things they can't in the demon realm (i.e. compassion, forgiveness, love, etc.), simply growing up (demon version of teen years/child hood? I'm graspin' at straws), figure out what they want to do with their lives, and get an idea of how they want to run the kingdom when it's their turn to rule. Also, so that it matches, I picture the demon realm being like the one in I Was Born as the Demon Lord's Daughter but my version of the demon realm is gigantic and has doors/portals to other worlds, with that being said Luo Binghe's, Hua Cheng's, and Wei Wuxian's canon stories take place in the 3 different versions/worlds of the human realm. To clarify, the demon realm has way more resentment/demon energy just floating around in the atmosphere and environment than the human realm allowing their demon blood to be more dominant while the human realm has the opposite affect.