Thank you for sharing your questions with the Wefugees Community!
In general, as an holder of a so-called Ausbildungsduldung, the authorities will likely grant you a residence permit for 2 years after you finished the (in most cases) 3 years of the Ausbildung. You will find the conditions and more details in this brochure (page 12).
One of the requirements for the permanent residence (Niederlassungserlaubnis) will be, that you were holding a residence permit for at least five years. So I am afraid that applying for the permanent residence immediately after you completed your Ausbildung will not be possible.
In order to receive an Ausbildungsduldung you have to provide documents that prove your identity or conclusively demonstrate why you could not organise such documents. Ideally, you would prove your identıty wıth your passport or a replacement document. Please check this thread, in which we discussed the requirements for the Ausbildungsduldung already.
I hope this helps and please get back to us if any clarification is needed.
All the best,