Thank your for reaching out to the community and sharing your question with us!
I've had a look and found this PDF document: https://www.institut-fuer-menschenrechte.de/fileadmin/user_upload/PDF-Dateien/Info-Flyer/Kopiervorlage_Geburtsurkunde_Deutsch.pdf
As it says, "The UN Convention on the Rights of the Child (UNCRC) is applicable law in Germany. Germany has committed itself to register every child immediately after birth (Art 7 Abs 1 UN-KRK [UNCRC]). Your child must be entered into the Birth Registry (Geburtenregister) in order for you to register your newborn child at the Federal Office for Migration and Refugees (BAMF) and so that it can have access to medical examinations and other services in Germany. Your child will need a birth certificate (Geburtsurkunde) or an extract from the Birth Registry (Auszug aus dem Geburtenregister) later in life as well, to enable public authorities to verify its data."
What that means is that it is essential to go to the Standesamt of the Bezirk (district) where you gave birth and register the child. Even if the asylum procedure is still in progress. The PDF further explains how to register your child, so it's worth it to have a look.
Here's another source I could find by the BAMF: http://www.bamf.de/EN/Fluechtlingsschutz/FamilienasylFamiliennachzug/familienasyl-familiennachzug-node.html
It states: "When a child is born in Germany after the parents have filed an asylum application, under certain conditions, the legislature provides the possibility of a separate asylum procedure in order to protect the child. For the latter, the parents, at least one of which must still be undergoing the asylum procedure, or the immigration authority, inform the Federal Office of the birth. The asylum application is thus automatically regarded as having been filed in the interest of the newborn child."
Regarding your last question, which benefits a child receives in Germany, here's another link by the BAMF on that specific topic: http://www.bamf.de/EN/Willkommen/KinderFamilie/Kindergeld/kindergeld-node.html
I hope these links and documents help answering your questions. Please do feel free to reach out to us again if you have any more questions!
Good luck with everything, all the best,