I have all level 3 tech categories, am at max population. Is there anything I can do to make the Golden Child be born? Are puzzles 10 and 11 both related to the GC?

Making more babies is the only thing that will produce the Golden Child, but you'll have to wait for some older villagers to 'move to the graveyard' before you'll have room in your village. Yes, both puzzles involve the GC.