Why? Because she cannot constitutionally fill the vacancy caused by the death, resignation, impeachment and removal of any US president under whom she served.
Correct. That is the author's hypothosis: a woman can't be VP "because she cannot constitutionally fill the vacancy caused by the death, resignation, impeachment and removal of any US president under whom she served". Once again, this has nothing to do with First Ladies or cabinet members. Forget that Hillary ever was the wife of a President or served in any federal capacity. She is a woman, and that is the author's only point.
No, it's not - otherwise, the author woult not have included the phrase "under whom she served." If a woman could not serve as president because of her gender, it wouldn't matter a whit whether she had served under a president or not.
In fact, for precisely the same reason that Bill Clinton can't serve as vice president, neither can Hillary.
Considering that, according to sworn testimony from Monica Lewinsky that Bill Clinton is a male, and according to the occasionally questionable claim that Hillary Clinton is female, their gender does not come into play according to the phrase "precisely the same reason."