1) To quote EE Vol 1: "Moogle reproduction in particular is a subject of enduring mystery, with any direct inquiries into the matter being casually deflected or simply ignored." We know that Moogles do identify with a particular gender, but how that correlates to their sex is unclear.

2) It's not exactly clear, but the implication is that the Shroud Moogles are a descendant branch of the Moogles from the Churning Mist. Good King Moggle Mog's story is old enough (possibly dating to before the First Umbral Era) that how much of it is legend and how much is fact remains murky, but the similarities between the two groups are striking enough to suggest that some of the Churning Mist Moogles departed to form the community in the Black Shroud.

3) Again drawing from EE, Moogle poms actually change color depending on mood, so they aren't a static color.