Unsure: the other answer to your questions about rights if the DNA father shows up is yes! you do.
The family law courts (civil law) deal with these matters.
If you are raising the child with your partner (childs mum) and have a stable, committed, non volatile home the courts are very very unlikely to give the bio father custody of the child. What they will consider is why the bio father has now come back on the scene, what he wants in seeing the child, what he is like as a person, who he lives with, where he live, what his background is like, what kind of environment is he going to take the child too etc. It is likely that the best case senario would be the bio father gets acees to see the child. Also depending on the age of the child the courts will ask the child what they want.
If the bio father gets access, is will likely be limited and supervised in the first instance. This is because the child will be going with a stranger otherwise.
From a professional backgroun I would suggest that most bio fathers who leave before the baby is born, are unlikely to proceed with court proceedings to see the child and when they do, most only stick at it until the novelty wears off or the child starts asking for money, treats, clothes or when the baby becomes less of a novelty.
As the bio father will not be on the birth cert, he had NO Parental Responsibility. So he can't just turn up and take the child, that would be kidnap!. ALso if he does apply for PR the courts HAVE to seek the views and opinions of all others who already hold PR over this child.
Hope this helps.