thank you for sharing your question with the wefugee community. Let me specify the answer given by Charington 26 a bit.
1. It's possible to do the custody declaration (Sorgerechtserklärung) and the aknowledgement of paternity (Vaterschaftsanerkennung) at the notary, but... it costs money at the notary and Jugendamt is doing it for free.
5. Yes, Sorgerechtserklärung and Vaterschaftsanerkennung can be done before the baby is born.
2,3,4. Vaterschaftsanerkennung und Sorgerechtserklärung can be done without a passport. The authoity will then state in the document that the identity of the parent is not proven. If you hand in a passport and a birth certificate Jugendamt/Notar will not write that passage. If you hand in the declaration to Ausländeramt later, without the passage ("Indentität beruht auf eigenen Angaben" oder "Identität nicht nachgewiesen" they will probably know that you were in possession of the stated documents.
I hope my answers were helpful and I would be very happy if you would contact me again with further questions. I'm a consultant with mbeon and I‘ll be glad to help you over a longer period of time. Mbeon is an online advice for migrants via the mbeon app, which you can install on your smartphone for free. Download it from App Store or Google Play Store. You can find me in the mbeon app and get confidential individual advice in a secure virtual space without other people reading our conversation. Further information: www.mbeon.de.