Yes, IIT BHU is an autonomous body, just like any other IITs. But certain powers are vested in BHU.
IIT BHU shares the same campus as IIT BHU. BHU and IIT BHU shares the same auditorium, badminton-squash court, swimming pool. Although IIT BHU has separate hostels, grounds, library, gym, etc. Also, IIT BHU organizes separate placements, fests, and other events.
Yes, IIT(BHU) is completely an autonomous body and actually all the IITs are autonomous bodies. IIT BHU was earlier the part of BHU University. It was converted to IIT in June 2012. Earlier it was known as IT-BHU. Before 2012, it came under the BHU administration but after being converted to IIT now it has all the rights to take decisions on its own. Academics, Courses, Campus Placements, Appointment of Director is done as per the IIT Rules and norms. But IIT BHU is not completely separated from BHU. It enjoys all the facilities of BHU such as OPD and medical Facilities, Playgrounds, Stadiums, etc. Also, there is a common E-Library where students from both the Institutes can access all the E-Books, Journals, and Resources.