Where is a hip arthroplasty usually done? At a hospital?

Usually. The hospital is the typical venue. There are centers for out patient surgery, but these are fairly rare.
Yes. Total hip arthroplasty is done in an or under the cleanest conditions possible. Anesthesia is either general, spinal or epidural. You will stay in the hospital for pain control and physical therapy, which may require stay at a rehab unit after hospital discharge.