Hello @Adi
Nice to hear from you again.
The Ausbildungsduldung allows you to stay in Germany for the duration of your Ausbildung (3 years). Afterwards you have six months to find a job in a related field. If you succeed you can stay for at least another 2 years doing your job. I don't see a reason why you shouldn't be able to live together with your girlfriend throughout this time.
Best regards,
Thorgen