Warning: 5th paragraph mentions puke, but not in detail
Danny was adopted by Bruce at fifteen, before Bruce had adopted Dick. Bruce knows about Danny's powers, but Danny doesn't want to be a Vigilante, he does step in when it comes to ghosts, and curses, and if Batman really needs his help, but thats it, and honestly Bruce prefers that.
When Bruce starts adopting more kids, its Danny who helps with some emotional regulation between Bruce and the boys, since Bruce his so pants at it. Danny is seen in the house as a bright sunshiny beacon who has all the answers, all his brothers are just (platonically) smitten, so nobody minds when its obvious that Bruce starts preparing Danny to lead the company one day, it makes sense hes the oldest after all and Bruce didn't know he had any bio kids at the time.
When Damian does come to the house, he has beef with Danny rather than Tim, since Danny was more obviously the heir in the eyes of the media. Danny thought he was adorable and happily let Damian try to kill him whenever he wanted, it only comes to a stop when one of Damian's attempts almost injures Dick, and Danny finally gets angry. The long scolding, followed with Danny doing his thing as an Emotional sensor, eventually gets talked down and Damian slowly starts actually trying to be a member of the family, he still sees himself above everyone as the blood son, but sits a start.
One of Danny's main roles, is taking care of the others when they're sick. Danny can get sick, hes not immune to illness, but the ectoplasm in his system is very good at killing viruses before they can get started. So, when someone is sick, Danny nurses them back to health, even Alfred who tries everytime to insist hes fine, gets fretted over, if not babied, by Danny.
That's why, when danny actually manages to get sick, rare it may be, its all hands on deck. When Danny is sick, he gets sick-sick, no tiny sniffles and the chills, he gets high fevers, horrendous wet coughs, and he pukes his guts out. He also gets very needy, if not babyish. Not as if he acts like a litteral baby, but his fever makes it to where he'll want Bruce to cuddle with him, and he wants his hair played with, and he'll cry at the drop of a hat while sobbing out apologies. Its such a stark difference to how he normal is, that the first time any of the Wayne's saw him like that, they were petrified, sure he was dying all the way.
Of course they did everything they could for Danny, he always did the same for them, so they worked together until his sickness would pass. Bruce, Dick, and often times Cass, cuddled Danny, and played with his hair like he asked, while Tim switched between looking at home remedies, because most medicine didn't work on Danny, and doing Danny's portion of work at WE, Steph leaning over his shoulder to add her two-sence. Damian would paint beautiful night time landscapes in the room, right in front of Danny's bed so Danny could see the stars whenever he finally woke up, while Jason lounged in a chair across the room, reading a book out loud so it wasn't too quiet, something Danny had admitted hating since he was so used to some sort of noise. Duke would sometimes use his status as a meta to buy pain killers and meds for Danny, even if they only helped for 30 minutes. Alfred of course was usually fretting around the kitchen, trying to find something Danny would be able to keep down.
Eventually, usually sometime between 3 days to a week, Danny's fever would break and hed slowly become more lucid. He'd blush and apologize for the trouble, and all of them would quickly assure him that he wasn't "trouble", or "a bother", he was their family, their loved one, and he needed help, so they helped him, and then Danny would usually cry.