i totally agree with litlehud. your baby should have her father's last name and you should be legally married, even if you don't choose to live together.
there are too many uncertainties in life for each of you not to protect yourselves and your baby. the time may come when you need the support of your 'other' and you must be in a position to get it. you never know what a family member can step in and demand or refuse.
also, as a single mother, you must never put yourself in the position of having your 'other' be able to say you're unfit. even though you've done nothing wrong and are not too young, you never know what the future holds. so take care of yourself as well as your baby.