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Although certain support positions within the Foreign Service do require a degree, "core" diplomatic officers do not. Want to be a diplomat? Again, just pass the test.
The Constitution explicitly states that he or she must be a natural-born citizen of the United States; naturalization won't do. State election boards generally take the word of the political parties, but as has been shown in recent years eligibility can be challenged in court -- most often by racist shitheels upset about the skin tone of the candidate in question.
A qualifying birth certificate is usually the path taken here.
So yeah, you don't have to show it unless challenged, but if you can't show it you likely WILL be challenged, and that could be a problem.
Fun fact: Trump rode birtherism all the way to the presidency, then never showed his own birth certificate.
That said, the law at the time was that if your father wasn’t a citizen, then your mother had to be a citizen AND had to have been living in the US for the previous 5 years.
Dumb law IMO but it was the law and was raised by a lot of birthers.
(Not that facts or logic matter to birthers, but anyway)
I think you misread the law, if she lived outside the country for two years then she wouldn't have spent the last five years in the US.
I think at this point we all have to acknowledge the retarded genius of Donald Trump, the way he always got people to talk about what he wanted them to talk about. Discussions like this legitimize false statements, and were part of what legitimized Trump himself.