I timeline shifted her a little, at this point she's 16, so technically starting a few years older than her male counterpart did as Robin, but the rest of the timeline would match up in the order of how things happened. Rayan's parents were killed just a few weeks before showing up and Bruce Wayne's taken her in as a favor for Gordon, since the kid's a witness to it Gotham PD felt she'd be safer in Gotham than with Haley's Circus.
Rayan agreed to stay, only because she figured the killer was local and wants to find him. She knows that Batman's in the city so is trying to find the man she only heard, without drawing the Bat's attention, hoping she can give the details to the police so she can go back home to the circus. Staying in one place too long makes her nervous after all.

Rayan's life has almost always been at least 40 feet above the ground. Since she was five, without a safety net. Living like that you either embrace fear, or you push it aside until you are firmly on tera firma again. Rayan; embraces it, the fear of falling is more comfortable to her than any blanket was as a child. Though it's a little more difficult to find that embrace living in a mansion in Gotham.
She hasn't been there long though and despite not seeing her host too often, his butler's always trying to chase her down from the roof, banisters or any other high surface she can find to perch at, much to his dismay. She does make up for it by bringing life into a home, where it hasn't been in years so Alfred doesn't complain too much.
When she's not annoying him or entertaining Bruce with her antics (Although she isn't aware of that bit), she's walking through the crime ridden streets trying to find a man that she only heard. Rayan knows that finding him is the only way she can return to the circus she loves, if he's in jail she won't have to worry about him coming after her and maybe endangering others in an attempt to take her out.

Rayan will probably spend a few days trying to figure out a way home, once she figures out she's stuck there she'll be trying to keep herself together, after all she can't hunt for Zucco trapped in a hotel. She'll likely try to distract herself by trying to help others. So I can see her trying to get along with everyone and trying to figure out the place. Once it really sets in they're stuck there, you can count on her to try to boost everyone's moral as long as she can, stuffing her own emotions down. I can see her needing someone to basically snap her out of that though. I doubt it would change her attitude, but a distraction for this pre-robin would probably be good. And maybe someone older to act as a mentor figure to her, since she will be lacking Bruce to do that.

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