Could you tell us more about your version of batman? Maybe the batfam too? Boop!
I can try! To be honest I don't have a ton to say, im not the most knowledgeable when it comes to the Lore(tm) and I'm mostly making stuff up with each drawing I do, whatever makes the initial concept or joke work.
So what I have is this:
He's not physically powerful but he has a lot of speed, flexibility and endurance and mostly focuses on avoiding fights and being a good detective, and when a fight is unavoidable he leans a lot on technology and being a pragmatic fighter so he can take people down quick with minimal risk. I don't know much about martial arts but his fighting style is probably similar to Akido?
The Joker is less of a huge villain, I have nothing against the character I just feel like he gets leaned on too much in batman stories, like an overexposure thing. He was a gangster with a gimmick who become a problem every once in a while but he was never Batman's arch nemesis. (That would be two-face, they got the most personal history). After the Jason Todd of it all Bruce stuck him down a stinky hole and no one liked him enough to help him get out.
There's not actually a huge age gap between Bruce and Dick, only about ten years, and the father/son relationship is getting slightly more awkward as time goes on. Like it's one thing for a 22 year old telling a twelve year old what to do, pretty different when it's a 35 year old trying to lecture a 25 year old. Dick respects Bruce immensely but he's getting frustrated with Bruce not seeing them as peers yet, and it's part of why Dick has physically distanced himself. (Some familial relationships improve so much when you live in different cities)
He's in an on-and-off again relationship with Selina, who has her own apartment but splits her time between it and the Wayne manor when they're on, and sometimes even off (Alfred usually let's her in regardless, when she feels like using the front door). They're just two very independent people who do love each other but every so often need their space and don't know how to communicate that constructively.
What else what else... Babs is Oracle but her spine was injured in a different way, probably while kicking ass and saving lives.
Bruce is better friends with Diana than he is Clark because Clark has a little bit of a country chip on his shoulder when it comes to wealthy property owners and Bruce isn't jazzed about the press. Its getting better with time though!
Bruce is very good at masking (I mean, clearly, he has to fool everyone with Brucie after all) but his relaxed affect is very blunt and not outwardly emotional. This does not mean he's always brooding or overly serious, he just has resting bitch face and his sense of humor is very dry. His family can read him pretty well but most other people just assume he's perpetually pissed.
Uuuuh thats all I can think of off the top of my head, hope you like it!
Also boop