Now if the courts refuse to hear the case(s), as they did with Obama, then he gets a pass and is deemed eligible. However, it also means that the question of what the original intent of "natural born" actually means, remains unresolved.
Had Obama not come forth with an Hawaiian birth certificate, fake as it was, would the courts have remained silent on the subject? I suggest that the would not have. Of course we know that Cruz was not born in the United States. Will this fact compel the courts to take the case? At this point no one knows, which is why the challenges will be forth coming without a doubt. It is not a settled case as Ted Cruz and others currently claim.
I don't think the courts will take the case, and I think they don't have to. The doubt is enough to sink Cruz. There is no need for a definitive answer. If the people have doubts, then the candidate is damaged goods.
Obama's case is different. The anchor baby cases are different. The public more or less accepts that, while not liking it, and the courts are biased to liberally grant rights.
As to "which parent" controls, either. For Cruz, citizenship and residence of the mother matter, and the father is irrelevant. That is a statutory issue, and delivers naturalized citizenship.
At the time of the founding, as you notice, citizenship of a couple tracked citizenship of the father, so one could figure out NBC by either looking at the couple, or at the father.
These days, with the family structure disintegrating, who knows. I think it would take a few hypothetical examples to figure out a "just" rule. Intuitively, I'd look at the parent who raised the child.