Yes, your residence card (Aufenthaltstitel) allows you to travel within Germany, but does not allow you to cross the national border.
You are, however, required to show and/or hand out your passport and/or residence permit to an authorized official such as
police and customs officer
government employees, such as foreigners and registration officials
Residents of the European Union are not eligible for VAT tax refunds when taking goods outside the European Union.
Since you are not a visitor (i.e. no entry stamp based on the 90/180 days rule), but a resident (you have a residence permit), customs will see that you are not eligible after a quick look at your passport.
You should, however, look into the ...