Thank you for sharing your questions with the Wefugee Community!
In general, the Ausländerbehörde will ask your girlfriend to get a national passport from the Embassy of her country of origin. However, if she can prove that her attempts were unsuccessful or there are any other important reasons which would prevent her from applying for a national passport, the Ausländerbehörde might also issue her a "Reiseausweis" - the so-called grey passport. One requirement is, that she holds a valid residence permit (Aufenthaltserlaubnis), which I guess will be the case, as she got a deportation ban.
Regarding your second question:
Your child has a residence permit according to §33 as it is born is Germany and, as I assume, as both parents hold a residence permit (it derives from their's). As far as I know, as long as both parents or the person, who holds the personal custody (who is the primary carer), have/has a valid "Aufenthaltserlaubnis" oder "Niederlassungserlaubnis", your child will also get extensions of it's residence permit.
I hope this will help. Otherwise, do not hesitate to get back to us with further questions.
All the best,