Hi @twilight_
I agree with the view of the social worker you spoke with. If a person has a Wohnsitzauflage and finds an Ausbildung, then that person is allowed to move. If they are married, their spouse can also move and likewise if they have children. It would only be possible for your mother to move with you (and your brother if he is under 18) if you yourself were still under 18.
You can see the rules around this in paragraph 12a of the immigration law (Aufenthaltsgesetz):
https://***.gesetze-im-internet.de/englisch_aufenthg/englisch_aufenthg.html#p0218
Section 5 covers how a person can have the Wohnsitzauflage removed. If you **** at part two of that section, you’ll see that it does mention the possibility to allow a move in order to prevent a particular hardship. It does not clearly define what is considered a difficult enough situation for this. The authorities will decide on a case by case basis. If your family members wish to apply for removal of the Wohnsitzauflage based on the hardship rule, then they have to make a strong argument. Especially important to mention are any medical needs your family members may have that staying together as a family unit can help with, or any other support you and your family members offer each other on a regular basis.
The application is called ‘Antrag auf Aufhebung der Wohnsitzauflage’. You apply with the Ausländerbehörde where you are living now and they will forward it to the Ausländerbehörde of the place you wish to move. You send proof of your Ausbildung with the application. Some authorities have specific forms to make this application. Otherwise you can just write it yourself. This site has a good sample application you can use, it’s in the section ‘Unterbringung, Umverteilung, Wohnsitzauflagen’:
https://***.fluechtlingsrat-thr.de/arbeitshilfen/antragshilfen
As you mention doing a schülische Ausbildung without a Gehalt, you may be entitled the financial benefit BaFög (in this case often called Schüler-Befög):
https://***.arbeitsagentur.de/bildung/ausbildung/bafoeg-und-kindergeld
Best of luck with the Ausbildung!
Éanna