Nearly all U.S. Supreme Court cases are appeals from other courts (federal or state) or requests for a "writ of certiorari." The Supreme Court has "original jurisdiction" over a few minor but significant disputes under the Constitution (Article III, Section 2).
Yes, Federal courts may exercise original jurisdiction or Appellate jurisdiction such that the court has a right to hear the case first.
The Supreme court is the first and the only court assigned in the constitution to hear a case.
