thank you for your question.
As for your question:
- yes the office is the same Jugendamt for both of you:
- due to the child and more important shared custody you will get an Aufenthaltserlaubnis for Germany;
- the duration depends on the fact how fast you will have all the documents such as Vaterschaftsanerkennung, Sorgerechterklärung and your Identity documents.
I hope this assist you, if further question arise please do not hesitate to contact us.