Padjal are apparently born looking like normal babies and manifest the horns at some point later on (it's not clear when they appear, iirc).
As to the rest of your questions: we don't know. There are a lot of holes. But, logically, if the Elementals are choosing unborn babies, the babies probably aren't given a real yea or nay in the matter. As for how the Gridania government reacts, your guess is as good as mine. That's never been spoken of in quest text, as far as I'm aware. Perhaps Sounsyy will grace us with a lore drop.
As to the rest of your questions: we don't know. There are a lot of holes. But, logically, if the Elementals are choosing unborn babies, the babies probably aren't given a real yea or nay in the matter. As for how the Gridania government reacts, your guess is as good as mine. That's never been spoken of in quest text, as far as I'm aware. Perhaps Sounsyy will grace us with a lore drop.