Yes. Kids are welcome. I've got a video of the get-together and there's children, even tots, there. It's not a throw-down drunk thing, but a nice, relaxing, socialble time before the evening meal. Hors derves are passed around as well as mixed drinks, non-alcohol ones as well.
As others have stated, kids are invited and they have nice Hor devers passed around as well as non-alcoholic drinks. They may find it a bit boring but what the heck, it is a great learning experiance for them.
No, you don't have to be a past guest for the Captain's Cocktail. The Captain's Cocktail Party is for all guests the night of the Captain's Dinner (formal night). Then there is what's called a Past Guest Party for past guest cruisers only. If you have sailed Carnival before and then you take a 5 day or longer cruise, you get invited to the Past Guest Party.