Federalists like Alexander Hamilton and other Framers did discuss having the US Supreme Court handle the impeachment process. They believed as Hamilton did that the Court should not be involved. Actually, Hamilton wrote in the Federalist Papers, that impeachment was more than just an issue regarding the breaking of a law. Hamilton wrote that impeachment was primarily a political process to engage the interest of the entire US community.
"The United States Constitution is the supreme law of the land." -A+
No. Alexander Hamilton was the first Secretary of the Treasury, and never sat on the Supreme Court. The person you're thinking of is fourth Chief Justice John Marshall.
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Yes; impeachment is the only way a justice can be removed involuntarily.
No. While a select committee can recommend that articles of impeachment be brought, it cannot bring charges (articles of impeachment) or conduct the impeachment trial. Articles of Impeachment are brought by the full House of Representatives, and the impeachment trial is conducted by the Senate with the Chief Justice of the Supreme Court presiding.