Welcome to the Wefugees Community and thank you for reaching out to us.
To answer your central question: Yes. In general, people are able to apply for a "Ausbildungsduldung" after the rejection of the asylum application and while holding a Duldung. However, as far as I know, these applicants are required to hold a Duldung for at least three months before being able to obtain the "Ausbildungsduldung". I tried to answer a similar question here before.
Regarding your question number 2: The BAMF will not ask for a vocational training (Ausbildung) contract. This authority is responsible to assess your asylum application and not to issue a residence permit or Aubildungsduldung in your case. For this, you will need to deal with the immigration office (Ausländerbehörde).
I saw that you opened a few other threads with very complicated questions. I am afraid that it is hardly possible to provide such detailed legal advice on this public platform. If the communication with your lawyer is not sufficient, I would highly recommend you to get some individual support in a refugee counselling office, for instance. Please let us know if you need help in finding such an organisation - you may also use this search engine though.
All the best,