My son (11), has a DS, but to be honest, he hardly ever uses a carrying case. He pretty much only uses his DS at home, and in the car, so he really doesn't need one. I suppose, though, that if your children might be using them other places, a carrying case might be a good idea. I would think that a smaller one might be easier to tote around.
As for games, anything that has Mario in it is sure to be a winner.
Here's a couple of resources that can help:
Gamespot -
www.gamespot.com - they have a good write up on every single DS game available (as well as GameBoy Advanced games, which also work in the DS). As well as screen shots, and user reviews.
Youtube -
www.youtube.com - here you can watch an actual video of pretty much every DS and gameboy game out there. Some times seeing screen shots and reading reviews isn't quite enough, so an actual video can be a huge help.
Amazon -
www.amazon.com - you can find literally hundreds of used DS and gameboy games for really low prices (under 10 dollars). Yes, they are used, but normally kids outplay a game fast enough so that the games are still in great shape.
Gamestop -
www.gamestop.com - They are also known as Electronics Boutique. They sell hundreds of new games for the DS (you probably have a couple of these stores in your home town).
Unfortunately only you really know which games your kids will like the best (for example many parents love the Sims games, but my son doesn't like them at all), so the best thing to do is to check out which games you think they might like before buying them.
I hope this helps a bit.