The name of a German national is governed by German law, even if his or her name is shown differently on a foreign birth certificate.
The name of a German national does not change automatically by marriage alone but only by Name declaration.
German law does not require the registration of a marriage concluded abroad of a German national at a German authority. However in certain cases it is recommended to apply for a German marriage certificate.
If you want to apply for a German birth certificate for your child you have to register the birth in Germany.