Well they do gloss over the whole King's Blood thing in the show. In the books, anyone who calls themselves a King is a King, and Mance was King Beyond the Wall, so his baby had King's Blood. It's the same way he's probably going to sacrifice Theon, because Balon called himself a King as well. Told myself I was gonna…

I don't know if we ever find out if it actually happened, but Gilly was certainly upset.

Someone correct me if I'm wrong, but in the books, doesn't Melisandre
burn a baby? Doesn't she burn Gilly's baby (that Jon passes off as
Mance's and then he sends Mance's real baby off with Gilly and Sam to
Oldtown)? I might be misremembering.