The President appoints federal judges, including Supreme Court justices, with the advice and consent of the Senate. That means the Senate must approve the President's nomination by a simple majority vote before the appointment process can be completed.
Any Article III (constitutional) court (or equivalent state court) can declare a law unconstitutional if the law is part of a case they're trying or reviewing; however, the government would probably appeal the case all the way to the Supreme Court to get a definitive answer.
The US Supreme Court is the ultimate arbiter of constitutionality, and has final authority over questions of constitutionality.
Article III Courts
U.S. Court of International Trade
Executive
political party conventions