Was it formerly illegal for a woman to be president of the United States? Was there some statute or other law a century ago that said only males could serve as president? If so, when did that change (if in fact it did)?
The language argument about the constitution is that the Constitution uses the pronoun "he" in referring to the president – they would not have used the construction "he or she", or "s/he". Article I also uses "he" to refer to qualifications of representatives and senators (residency, age). Then in creating the office of predident pro tempore of the Senate, it uses "he" to refer to a Vice President who is exercising the office of President of the United States. In Article IV Section II it states: A person charged in any state with treason, felony, or other crime, who shall flee from justice, and be found in another state, shall on demand of the executive authority of the state from which he fled, be delivered up, to be removed to the state having jurisdiction of the crime. Thus women cannot be extradited, if the "'he' means male" theory prevails. In 1872, Victoria Woodhull ran for president, and while it is true there were questions about her eligibility, that was because of her age rather than sex. Belva Lockwood ran for president in 1884 and 1888. This is not strong enough to constitute proof that being female was not a bar to holding the office, but it is indicative of that conclusion. The strongest evidence that the use of "he" in the Constitution had no significance is the fact that Jeanette Rankin was elected US Representative from Montana in 1916, 2 years before Congress approved the 19th Amendment. Since the only way to impose any restrictions on the presidency is via the Constitution (i.e. Congress can't pass a law redefining the qualification for office), and since the use of "he" has never been strictly interpreted to preclude female senators and representatives, we have to conclude that there never has been a prohibition against a female president.
Everything is allowed unless the law says it isn’t Common law systems like the USA are ‘exceptions based’ - the law permits everything except what it prohibits. So, your question is backwards - rather than looking for laws that allow it, you need to look for laws that prohibit, restrict or regulate it. There are laws that regulate this but none that prohibit it.
Maybe s158 of the Equality Act 2010 allows proportionate action to redress disadvantage, different needs or disproportionately low participation by people with a protected characteristic. For example, woman make up 16.5% of the UK’s engineers. Given they make up more than 50% of the population one could reasonably come to the conclusion that this is disproportionately low. If this was an engineering scholarship, this would fall within the exemption. In contrast, woman are 52% of UK lawyers - not a disproportionate number for either gender and not engaging the exemption. However, female judges are only 39%, arguably disproportionate, so if the course is a post-graduate one preparing people to enter the judiciary, that’s probably ok. Women are 75.5% of teachers, so a scholarship for male students in that profession is ok too.
Yes. US Constitution, at Article two, Section two, provides that the President has the power to nominate the justices and appointments are made with the advice and consent of the Senate. There is nothing restricting the President from doing this on any particular day from beginning to end of his term, and also no restriction on when the Senate may give their advice and consent.
Gaining UK citizenship at birth would not count, the law stipulates "after having attained the age of 18 years", so it is explicitly ruled out. (Also, gaining UK citizenship at birth involves no formal declaration.) The scout promise would not count. It doesn't involve swearing allegiance to the Queen, only promising to "do your duty to the Queen"; and an American could argue they have no duty to the Queen. (Also, you might well not have repeated it after 18.) On the face of it, the Oath of Allegiance for the OTC would count - it looks pretty much exactly what the law-makers had in mind. On the other hand, the oath is almost exactly the same as that taken by Members of Parliament, and Boris Johnson took it first in 2001 - but didn't renounce his American citizenship until 2017. (Perhaps it wasn't until 2017 that he finally drew the American authorities attention to the fact he had taken the oath.)
Congress has the power to propose amendments, but not to enact them. Amendments are only enacted once they're ratified by 3/4ths of the state legislatures. And yes, there's no reason to think it would be unconstitutional for 2/3rds of each house of Congress plus 3/4ths of the state legislatures to make fundamental changes to the Constitution like eliminating other branches of government. The only limit on amendments that's still in effect is that states can't be deprived of equal suffrage in the Senate without their consent.
Does the unitary executive allow the President of the United States to suspend the law at his discretion for purposes of national security? No. The unitary executive theory pertains to independent agency autonomy, not to the authority of the executive branch to disregard statutes.
In the US, Congress may pass an act, and this creates one kind of law (if it is promulgated: signed, ignored, or re-passed with a super-majority). Some of those acts direct the executive branch to do things, and pursuant to that act, a regulation is promulgated. Together with case law, the whole thing is "law". A bill (in the House, or the Senate) may result in an act being passed by both houses. It may go through a number of drafts between the point when it is first introduced and the time it becomes an act.