[so_real]

I'll bite.

What law lays out the procedure by which a candidate establishes his qualifications for president?

None. That's why the SCOTUS is involved today. Prior to this election, common sense largely prevailed. Common sense has left the building.

What agency is tasked with checking the qualifications and how is it done?

Department of State for each state in the union as deemed acceptable by the constituency of the state and overseen by the Office of the Secretary.

Bottom line ... the POTUS cannot serve two masters. Having dual citizenship precludes 'natural born' status. Renouncing foreign citizenship, or allowing foreign citizenship to lapse, does not instate a natural born status. It is that simple.

[Non-Sequitur]

Department of State for each state in the union as deemed acceptable by the constituency of the state and overseen by the Office of the Secretary.

I've lived in Kansas for many years and I wasn't aware we had a State Department.