billing for physician services in nursing home

When a internist sees a patient admitted to a nursing home for short term rehab, aka skilled care, who should they bill. Is it the patient's medicare part b services or bill the nursing home directly?

If we're supposed to bill the nursing home, do they have their own fee schedules that can be negotiated?

And if we bill a nursing home, then why do we have to meet the medicare documentation requirements for e&m cpts codes?