To clarify, I was awarded a Fulbright Research Grant and the DAAD Study/Graduate Scholarship. I was pleasantly surprised to receive a 2nd email from the DAAD notifying me that I was also selected for a 2-month summer intensive course, despite not applying for anything of that nature. I have not heard anything about the Fulbright language course besides what's listed on the website, though I do know that you must be selected.
In terms of classes, I plan to enroll in German language courses at the university. I'm already affiliated with a lab/research group and was hoping to also obtain "student status," thus allowing me to take classes without having to pay tuition, or at most, paying a minimal fee.