As to the question of Cersei's kids, in the novel, it is actually made more clear. The book that Ned reads (the one Jon Arryn had gotten from Pycelle) gives him the clue when he reads about how EACH of the previous pairings of Baratheon and Lannister ended up producing a black haired child. It wasn't just that every Baratheon babe had black hair, but specifically that the ones from a union of Stag and Lion had the black hair.

As to the dragon eggs hatching, I got the impression that it was a combination of the intense heat of the funeral pyre, the death of Mirri Maz Duur, and the blood of the horse Dany had slaughtered. Of course, this doesn't really mean anything, since it is stated in the books that intense fire (along with deaths) had been tried before to hatch eggs, to no avail. PErsonally, I think it has more to do with Dany being a true "dragon" where her more recent ancestors weren't (which includes Aegon aka Egg from the Dunk & Egg stories). And too, there are other things effecting the world, like a ************ SPOILER WARNING **********************************************************************************************************************************************************************************************************************************************************************************************************************************
















big red comet appearing in the sky around the same time the book was ending (you don't read about the comet until the prologue of book two).

I personally think somehow the return (or first time arrival) of this comet has helped bring magic back into the world, allowing everything Dany did with Drogo's funeral pyre to hatch the eggs.