Hello Faga0206,
you mustn't write to BAMF yourself. Normally if you (parents) are asylumseekers in asylum procedure, inclusive court procedure after rejection, or if you have Duldung, BAMF will start an asylum procedure of your new born child themselves as soon as they have information about the birth from the Ausländerbehörde. That means BAMF will send you a letter (provided they have your proper current address) with the registration and the number (Aktenzeichen) of the asylum procedure of your child and they will ask you if you agree to this procedure or you wish to renounce it. And if you don't renounce, they will ask you to give asylum reasons or reasons for another kind of protection for your child. Sometimes children can have their own individual reasons which are not the same as the reasons which their parents gave before. Example: a new born girl from Somalia, Nigeria etc. who can become a victim of female genital mutilation in the parents' country of origin.
I wouldn't speed up the procedure of the child myself in your place because it prolongs the whole safe time in Germany for the family (the time you all can't be deported). If the child is rejected, you can make an appeal for him or her and there will be one more court procedure. It gives you more time for integration.
With social help for a child it mustn't be a problem at all. Even if you make no asylum procedure, the child can't be deported alone and would stay with Duldung, that means the same social money from Landratsamt. You must just inform them about the birth and give a paper from Krankenhaus, later you shoul care about papers for the child from Standesamt. But the child can only have a proper birth certificate if the identity of the parents is fully cleared and you have all necessary documents.
If you want to make Ausbildung, you must find an employer who will give you a contract for that. Normally you need at least B1 German, sometimes B2. For "Altenpflegehelfer" you need A2, but this is not a qualified Ausbildung.
Every Ausbildung is special, you should ask what minimum starting level they need.
About your school certificate you should contact a councelling office in your area which is responsible for the question of checking and accepting of foreign education documents in Germany. In Baden-Württemberg it is, for example, IKUBIZ, for other parts of Germany you must find out. Perhaps your local Bundesagentur für Arbeit has a corresponding councelling.