I took a quick look at the U.S. Code for any provisions regarding procedures for establishing qualifications to serve as president and could find nothing. I am also curious to know if the Supreme Court has ever construed Art. II, Sect. 1, fifth paragraph (I doubt it). Also, who would have standing to contest the qualifications of the winner of the electoral college, and who would have the burden of proof in a proceeding?
I am only going to respond to a very narrow part of this.
Article II, Sec. 1, Par 4 provides that a person who does not meet the test is not "eligible" to be President. Thus all of his official acts would be subject to challenge as void, one at a time, in any available District Court.