No. The US Supreme Court only exercises appellate jurisdiction over criminal cases originating in the state and federal court systems; they do not hold trials. The federal court of original jurisdiction (trial court) for a murder case would be the US District Court overseeing the territory in which the crime occurred.
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The Supreme Court is the most powerful federal court. The Courts of Appeal are the most powerful courts most litigants will ever reach (the Supreme Court only hears a tiny number of cases a year). The District Courts are the trial level courts.
I'm not sure what you're trying to ask, but there was no recent impeachment of US Supreme Court justices. The only justice ever impeached was Samuel Chase, in 1804. Chase was acquitted at his Senate trial in 1805 and continued serving on the Court until his death in 1811. For more information, see Related Questions, below.
Supreme Court justices have lifetime appointments--most of them retire, but they can stay on the bench until they die like William Rehnquist did. They can be impeached, but that's the only way to get rid of one. The only justice to have ever been impeached was Samuel Chase, who was acquitted. Samuel Chase was nominated to the Court by George Washington.
The power of the supreme court to declare a law "unconstitutional" is called Judicial Review. No part of the constitution actually grants this power explicitly to the Supreme Court, but the case of Marbury vs. Madison established this power and has been the accepted precedent for granting the Supreme Court the power of Judicial Review ever since.
William Taft was both the President of the United States (Executive branch) and a Supreme Court Justice (Judicial branch).