If the firewall is killing the Internet, you have something configured wrong, since that's not the idea behind a firewall!
Since you have a router, and you haven't opened any incoming ports, you should be pretty safe from outside threats. I'm not sure what is going on with your Norton Firewall, I don't use it. I installed Norton Internet Security once, but it was a resource hog, and setup was more complex than necessary.
My advice with kids is to give them their own computer, and make an image backup of it before you turn it over to them. When it goes down the drain with whatever they manage to do to it, restore the image backup and they start again!
