President's minimum age

[True Federalist (진정한 연방 주의자)]

Two cases from the Senate seem on point.  Joe Biden was 29 when elected in 1972, but was 30 by the time his term began. Rush Holt Sr. was 29 when elected in 1934 and was still 29 when his term began and did not take office until after his 30th birthday.  There may be some cases in the House as well, but I'd say those are enough to say that what matters is what age the President-elect is when taking the oath of office. As for exactly what day one reaches age 35, one will find both the common-law definition of the day before the 35th anniversary of birth and the now common statutory definition of the 35th anniversary itself used in Federal laws and regulations.  For this purpose, I would say the common law definition prevails unless there is statutory law addressing it.

As for the time zone issue, I'd say it's settled on when you're born.  Both me and my stepdaughter were born in the wee hours in the Eastern Time Zone.  Our birthdays would not suddenly become a day earlier if we moved to California.  There's no age-related issue under the law that I'm aware of where the exact minute is taken note of. (Indeed, the potential uncertainty for those such as us is likely why the common law tradition of having one's legal age go up the day before the anniversary of one's birth exists to begin with.)

[True Federalist (진정한 연방 주의자)]

Three Senators, including Henry Clay actually served in office before turning 30 (the last in 1818).  I've sometimes wondered in you could make a constitutional challenge against bills passed that one of them voted for.

Probably not. "Each House shall be the Judge of the Elections, Returns and Qualifications of its own Members". While in hindsight, Congress made errors in enforcing thr qualifications in the 19th century, that wouldn't make the the laws they passed invalid.