it is not possible to say anything about the kind of residence permit which the Ausländerbehörde wants to give you without knowing their reasons. Normally the residence permit because of German children is better, stronger, more difficult to take back from you. If your child is not German, there is no 100% guarantee that this child will keep the residence permit forever, and you depend on him or her. And you can have more problems with prolonging in some other cases. Perhaps the Ausländerbehörde has some special reasons for their decision which your lawyer can evaluate and give you an advice if it makes sense to accept it.
The decision of the court about the financial help can take much time, all courts have different timeframes. The lawyers who have many processes in the same court can sometimes have an idea but it always can be shorter or longer.
You must know that this money is not a present for you but more like a loan. If you start to earn enough in the next 4 years they can claim that you must pay back. Naturally if you are financially responsible for your children, you must really earn more than the special limit for all persons, only then they can claim for money.