The president has some influence over the supreme court, but only to an extent. As previously stated he nominates people to the supreme court, but this in turn is checked by the legislative function in the government so his influence is small here if anything. He does however have the power and right to openly support or criticise the court, thus to an extent influencing some of their decisions.
The President has the power to fill open judge positions by submitting nominees of his choice to the Senate. He can choose people who agree with him on constitutional questions.
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The Legislative branch checks the President's influence over the Supreme Court because nominations to the Court can only be appointed with the "advice and consent" (simple majority vote) of the Senate. If the President and majority in Senate represent different parties, this prevents the President from nominating someone whose ideology is too extreme.
The Vice-President does not preside at an impeachment of a President. The Constitution requires that the Chief Justice of the Supreme Court preside over an impeachment trial.
No. The US Supreme Court and International Court are unrelated and have jurisdiction over different types of cases.
How have the supreme court has changes
No. Only the President of the United States can do this. However, when a law is enacted, and then challenged in the court system, it may be appealed all the way up to the Supreme Court where it may then be ruled unconstitutional.