Registering a birth in Germany

Family registries ("Standesamt") are among the most inflexible and difficult to deal with authorities in Germany. You will not achieve anything unless you follow their procedures to the last dot.
After the birth of your child, the hospital (do not even consider to give birth at home - the formalities would be overbearing!) will issue a birth notice. You have to visit the family registry with that and a number of other documents, among them birth certificates and residence permits of both parents - all in originals, they do not accept copies (and neither missing documents).
A German birth certificate does not mean German citizenship for the child. For that, it would be required that one of the parents was German at the time of birth. But unlike almost all others acquiring German citizenship, those who get it by birth together with another citizenship can remain double citizens for life.