Dear @Em,
Welcome to the Wefugees Community and thank you for reaching out to us.
I am not a lawyer or an expert on family law. As far as I know, however, are the rights of a "life partner" (especially when one half of the couple is still married) very limited. Since you are asking for the privileged requirements for the naturalization process (getting the German citizenship) - they are only applicable to official spouses or registered same-sex partners of German citizens. For more information you may read the information on this official German authority website.
Regarding your second question: In Germany, if the parents are not married, the custody is always with the children's mother unless the father is officially recognizing the paternity and the parents together officially declare the custody. This is usually to be organized at an appointment with the responsible "Jugendamt" (Youth authority).
I hope this helps and please don't hesitate to get back to us with any further questions.
All the best,
Meike