Yikes, some of the staffie comments make me so sad. It is a shame that a bunch of idiots owning the breed have brought it to ruin. Like any dog, you shouldn't leave them unsupervised with children and I think it's important if any dog is going to be around young kids that you make sure you are clued up on its history, whatever the breed.
Also the comment about crossbreeds/mongrels needs to be taken with a pinch of salt IMO. Whilst I agree that any dog breed will come with tendencies to certain health issues, if you do your research and make sure you go to a reputable breeder who carries out the necessary health tests then you are taking a big step towards minimising heartache. In addition to this, accidental crosses or designer breeds won't mean that your dog is less likely to suffer for hereditary conditions - it just means that they could run the risk of inheriting poor genes from both breeds. Obviously I'm not talking about a full on mongrel here. (Sorry, went off on a bit of a tangent there!)
From the breeds you've listed, I've heard great things about border terriers with kids! Also if you want a breed that can run for miles in the heat maybe take a look at the Rhodesian Ridgeback.