Child Chinese Travel Document

1. Father has British Overseas Territories passport, mother has Chinese passport. Live in Bermuda (British Overseas territory). Can Washington or New York Chinese embassy process Chinese Travel Document for child, if parents plus child fly to either state to get this done? (These are the closest jurisdictions to Bermuda to have the Chinese Travel Document processed.)
2. Does the Chinese Travel Document allow child to enter and exit China multiple times to visit family on vacations?