Regarding my child's custody, can the mother take her away from me?

Ok lets back it up here-

You say there's no court documents right? That means there was never a custody hearing in regards to your daughter, all you have is a mutually agreed upon decision that the child would live with you.

Legally speaking here, because she your daughter was born in South Carolina/mother resides there the laws are very clear! If you/your ex were never married, your ex actually has legal custody of her, because of the fact neither of you went to court! What does this mean? It means that if your ex came tommorow to pick her up, she can, and there is nothing you can do about it, she would automatically go home with her mother. You would have to file a custody petetion with the courts to get legal custody of your daughter.

What you need to do is make this offiical and file for sole custody.

None of the people who answered you are lawyers. Asking readers for legal advice is like asking readers for advice on how to perform surgery. Please consult an attorney who deals with family law. 

not if you didnt do anything wrong

You have absolutely no legal ties to your child? Please have a DNA test done. The fact that her mother apparently gave her away does not establish you are the legal custodian. Get a DNA test.

Get your name on that birth certificate. Demand a DNA test from the court.

Yes. If you're not shown as her father/legal guardian, the mother can claim kidnapping. You need to have this resolved legally so that anvil isn't hanging over anyone's head. How the heck did you enroll her in school and get her medical care all this time??

Yes she can. You better make your custody official in court.