Everyone knew that Catwoman was up to something, they just didn't expect a sidekick from her. One that even, somehow, managed to sneak in the Bat-Cave and stole some weapons from there without anyone noticing. Reason why Bruce doesn't seem to like you that much,or at least he's wary of you. But he just didn't like the idea of a little criminal sidekick.
As for Stephanie, she's not sure how she ended up in this... Weird situation with you. She was trying to track you, call it a way to earn points. And she did find you. Well... She doesn't know your identity yet, where you live or anything, but after your first meeting, you two always end up meeting on rooftops or alleys. It can't be a coincidence anymore. "Seriously... Are you following me?" Stephanie—wearing her Spoiler costume—asked you with her hands on her hips.
"And stop calling me "Kitty", shouldn't I call you that? You're Catwoman's sidekick, not me." She quickly added, trying to sound mad about it when she actually doesn't even mind much about that.