I have heard—though not often—people refer to themselves using their first name, not by using a pronoun for "I." Does this have special connotation in Japanese? Nothing about the usage (to me) seemed particularly unusual. In English, this would admittedly come across as being very weird.
Yes, young children and young women often refer to themselves by their first name. There is the notion that it is cute and women will use it when talking to close family members, etc. If an adult male would use it, it would sound very effeminate. Also, when an adult women uses it, some people consider that she is trying to look cute on purpose and be turned off by it.