OP, I agree with the others that you seem to have good intentions, and I don't find them rude or creepy or weird. The only thing that might be a little "off" is if, when talking to your friend, you were really specific about what you envisioned happening, without it having any grounding in reality--we all daydream about what we would like to have happen, with friends or family or work or whatever, but at a certain point most of us know when it's just fantasy and not realistic. Like, if you were really frustrated with your boss, and you told your friend that you really WERE going to march into Boss's office and tell him off--instead of saying that you merely wanted to do that--your friend might be kind of horrified as well, and try to dissuade you from doing it.
Anyway, maybe your friend just got the impression you were REALLY going to contact this girl for the first time by offering to have her come live with you and pay all her bills. As opposed to having thoroughly thought out a lot of details and precautions already.
I have friends who have opened their homes to vaguely-connected people in the hopes of giving them a hand-up that they weren't receiving from anyone else. Personally I couldn't do it, I'm too distrustful of people. I think if you want to wade into this, it needs to be done slowly and pragmatically, with eyes wide open, and not blinded by sentiment or idealism (not saying you are, just in general). If you're just hearing her complain on Facebook, you are probably not getting the full story.
I would start by contacting her and seeing how she responds to an electronic relationship first, where you can ask questions and have longer conversations than just Facebook status updates. Maybe see how she feels about you corresponding with her parents or other nearby relatives as well, so you can get a broader picture of the situation. Move on to small gifts (<$50) and short visits. Then, mention the school nearby and see if she's interested in attending it, were money and distance not an option.
This is not all about protecting you, BTW; think about her, receiving an offer of room, board, tuition, and babysitting for FREE from a virtual stranger out of the blue--she will probably welcome the chance to get to know you better herself, before making the huge commitment to put herself and her child under your care.