Like it or not, the very document you site grants the Federal Government to enact laws. Article 1, Section 8, Clause 18.
It is up to the Supreme Court to decide whether or not a law is unconstitutional or not.
Yep. But it takes a Constitutional Amendment to change what is stipulated in the Constitution.
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Section 1 - The President
No person except a natural born Citizen, or a Citizen of the United States, at the time of the Adoption of this Constitution, shall be eligible to the Office of President; neither shall any Person be eligible to that Office who shall not have attained to the Age of thirty-five Years, and been fourteen Years a Resident within the United States.