What type of doctor best treats an adult patient with moderate cerebral palsy?

Rehab physician. Depends on the problem, but for overall cp-related care I would recommend a specialist in physical medicine and rehabilitation (pm&r). For seizures, a neurologist; for constipation and gerd, gi. For scoliosis or hip pain, orthopedics.
It depends. I am a pediatric orthopaedist, but still see adults with cp, as I have more experience with the usual issues than my adult colleagues do. If it is for arthritis or similar issues, an adult reconstruction specialist can help. I would call a pediatric orthopaedist in the area and ask them. If you are asking about general health issues, any primary care physician is well qualified.