The context doesn't change the definition, except that you didn't include all of it. After that part that says, "For the purposes of this case it is not necessary to solve these doubts," it says, "It is sufficient for everything we have now to consider that all children born of citizen parents within the jurisdiction are themselves citizens." They don't say that they can resolve the doubts, just that there's no reason to try as they have a defintion for which there is no doubt. If they didn't have such a definition, then they would have to figure out how to resolve an alternate definition.
The decision does not address Obama's birth circumstances and so can't be used to either prove or disprove his citizenship status.
Sorry, but you're wrong. This does address Obama's birth circumstance in that the definition for which there is no doubt precludes Obama from being a natural born citizen. What you don't seem to understand is that Minor v. Happersett was argued well after the ratification of the 14th amendment. The significance is that if the courts thought it redefined natural born citizenship, this would have been the perfect opportunity for them to say so. It would have been easy for them to say the 14th amendment solved whatever doubts they have, but instead, they tell us quite clearly that it does not, for the only definition of NBC with no doubt is defined OUTSIDE of the Constitution.
No, it doesn't. It merely says in this one set of circumstances there is no doubt. The judge goes on to recognize other possible birth circumstances without attempting to resolve the "doubt" because it is irrelevant to the case.