No, it's how he wants to do it, and the implication of "again" shows he believes it has been greater than it currently is (in contrast to all of the Democrats who've said it was never great) that make this a very different thing.
No, it implies it has been greater. What country is the country of his birth that would imply he should go there?
That's never happened. Never once has the question come up because, or simply because, Obama is biracial and, to be honest, I have no idea what race (nor do I care) what Omar is.
While the allegations against Obama had sufficient merit to investigate the issue, and the question is further confounded by the fact that Mr. Obama spent millions to defend against having to prove his citizenship vice just proving his citizenship, Occam's Razor suggests it is most likely he was born in HI (or, Seattle, where I believe his mother was living at the time) vice Kenya. I have no reason to believe otherwise.
As for Omar, a similar situation exists where she refuses to show who her siblings are in order to help alleviate the question, which is rather odd. Her first and third marriage are to the same guy, though the first time she was married she (characteristic to her faith and nation of origin) never filed that marriage with the government (kind of proves my point about homosexuals that they don't need the state to be married unless their religion is the state, doesn't it?). It is the marriage in between that is in question, and she's done nothing to throw the question back in the face of the accusers by showing how wrong they are. For such a feisty person, that seems strange. Combined with the accusations of fraudulent entry into the US, there's circumstantial evidence to suggest it is worthy of investigation. But, until such investigation is thoroughly complete (like the Mueller investigation was allowed to do), there's no reason to assume an illegal action is fact.
Note, neither of those unbiased explanations has a thing to do with the folks' race, unless you want to divide people and assume racism in EVERYTHING, which seems to be in vogue.