[Uncle Sham]

"What law gives the responsibility to what federal agency for examining candidate qualifications and ensuring they are correct? "

The Constitution is law. Twentieth Amendment, Section Three. The federal agency is "Congress". They are guilty of eligibility nullification.

[0.E.O]

The Constitution is law. Twentieth Amendment, Section Three.

Then who does the Constitution say must determine whether the President is eligible. It's not in the 20th Amendment itself.