One. The US Supreme Court is a separate entity from the other courts in the federal judicial system.
No. The US Supreme Court and International Court are unrelated and have jurisdiction over different types of cases.
The US Supreme Court serves as the highest appellate court for cases appealed under its federal question jurisdiction.
The US Supreme Court doesn't use a standard court reporter, as trial courts do. The position most analogous is the Reporter of Decisions, who is responsible for ensuring oral arguments are recorded...
The Supreme Court is a Federal Court.