They make you use a real sounding name, but plenty of people change it. My SD isn't using her real name and neither is a friend of mine. Another person only uses FB for her cat.
What you should do, IMO, is consider any picture "public" even if you make it private. And you should limit the pictures you post of your child, even without her info attached. I am very careful what I post of DD and people are starting to understand that I won't post their kids without permission and I will ask them not to post mine. I don't think you're in particular danger from one image, but I would be more mindful in the future of where, when and what based on this. We attend some big public events and DD is cute so occasionally she ends up online. I just do what I can to monitor when and where.
As for public record info...it's out there. Not to make you paranoid, but data mining is crazy what they can link together. We've gotten calls about my DH's ex's ex before - thanks to the spiderweb of info that's out there. But you can help this if you don't fill out every field on every form. FB won't crash if you don't enter a phone number. Maybe you can't use it on your phone, but you can still use it. Etc. I don't have my HS, college, job, etc. on FB. Nobody I am friends with needs to know that info outside of our real world friendship (I don't friend people I don't actually want to go to dinner with). But don't worry about every little field because if you have utilities, it's in someone's database somewhere.
I think your kid is cute, and I can see why you posted it and I'd at this point just consider it lesson learned that things intended for one audience don't always stay there. So be more selective and move on. I don't think she's in any more danger than any other cute little kid. When she gets old enough to FB, make sure she leaves off info like her grade and school. And since she's 4 now, right, you can start to talk to her about things like not going off with people who say you sent them, etc. A 4 yr old should pretty much always stay with an adult.