Hello @aaudi624 and welcome back at Wefugees!
I will try to answer some of your questions, but in general I would highly recommend you to get some individual counselling in a migration counselling office to elaborate on your options and the next steps to take. In Berlin you find numerous well experienced, free of charge and confidential counselling services. You may use this search engine to find the respective contact details.
If I see it correctly, my colleague was also providing information on some of your questions in one of your earlier opened threads here. Maybe you want to check his answer again (especially in light of your first and third question).
If you want to move to study, you have to submit an application to change the residence requirement (Wohnsitzauflage) at the immigration office. As far as I know, they will also involve the immigration office at the place where you want to move.
Regarding your status: Unfortunately, it will not change just because you are enrolled at the university. I am not a lawyer, but as far as I know, you would need to re-enter with the respective visa in order to get the residence permit according to section 16b of the German Residence Act (for students). Generally speaking though, the better you are qualified and integrate yourself in Germany, the better your prospects for residency will be in the long-run.
As far as I know, you may be able to receive Bafög now, since you are in Germany for more than 15 months (which is the minimum to be eligible).
I would really discuss with an expert in a migration counselling office whether it makes sense for you personally to switch to a training occupation (making an Ausblildung). This depends on many factors.
I hope that the information helps you a little further and please let us know if you struggle to find counselling services or if you have any other question.
All the best,
Meike