Just because a person is born on American soil does not make that person a citizen. (Take the children of diplomats, for example.)
"All persons born or naturalized in the United States, and subject to the jurisdiction thereof, are citizens of the United States and of the State wherein they reside." If you aren't here under diplomatic or some other kind of immunity, you're subject to the jurisdiction of the United States; see 83 US 36 and 112 US 94.
Does the fact of one of his parents being a British National confer British citizenship on him? Dual citizenship? Does it depend on the laws in effect at the time of his birth? How does that affect his eligibility?
Maybe; I'm not familiar with British citizenship law, but I imagine that without being born on British soil, application for citizenship under jus sanguinius would be required when he wanted to claim that citizenship. You can have dual citizenship in both the UK and the US. As it turns out, the Constitution only cares that you're a "natural born citizen," which clearly means that you're not a naturalized citizen. Being a citizen by jus soli or jus sanguinius means that you were born into citizenship (by location or by blood), which is about as "natural born" as you can be. Also, 169 US 649 would seem to indicate that he is indeed a citizen by the 14th amendment unless said parent happened to be working for the British government in an official capacity, which isn't the case.
If his mother became an Indonesian citizen, doesn't that mean he, as a minor, was also an Indonesian citizen? Doesn't he have to file a form during his 21st year asserting his birthright to American citizenship? (If he didn't, is he an illegal alien?) Did he attend Occidental College and Columbia as a foreign student? If so, how does that affect his eligibility?
Maybe; I'm not familiar with Indonesian citizenship law. However, in most countries, the mere act of your parents being naturalized doesn't have any effect on your citizenship, in much the same way that a child of a foreign national, born on US soil, doesn't immediately make his or her parents into citizens despite the rabid claims about "terror babies." As we've already established he's a citizen by jus soli, and US law assumes anyone born on US soil is a citizen unless a proper objection can be raised to the contrary (and in this case, that'd be that both of his parents were not subject to US jurisdiction at the time, or that the birth certificate is fake, and both of those objections have been disproven), no forms need to be filled out. I don't know where this "file a form during his 21st year" thing is coming from, since the only relevant form here to assert citizenship in the US is the notification of foreign birth, which is filed by the parents with the State Department after the birth in cases of jus sanguinius where the child is born outside the US.
I am bothered more by the fact that Obama and his groups have spent millions of dollars trying to suppress attempts to find out the facts, than I am by crazy people spreading doubts about where he was born.
Really? Because I think crazy people spreading doubts complicates the political discourse to no advantage and is essentially demeaning an institution and a person with no evidence. In my book, that's rather unethical. Would you be okay with people bringing up doubts here about your sanity, or your recent battles with drug abuse? See how easy it is to "spread doubts" that serve no purpose other than to engage in a cheap shot against someone with whom you disagree?