Another thought; our local pet store gives out trial size kibble for free, you might be able to see if your dog may like one kind more then another, chicken over salmon, or something of that kind. I suggest to stay away from most commercial products and get the food from a local pet store, they tend to be able to give lots of advise too. You might also reward him with quality kibble, one at the time for good behavior. I like to use a kong to engage my dogs attention a bit longer and gobble the food down a bit slower.
Wow, what you describe is scary and demands immediate attention! I agree, no sleeping with you, no way! Not even the slightest signs of aggression towards the kids!. No pulling back the lips even, there has to be a consequence! This is most important and really concerns me. The dog should not dictate his food. Try mixing in better food with the one he is used to and gradually switch. Or else let the dog get hungry enough he will eat whatever is put in front of him. Cesar Milan's number one tool is walks on heel, so that your dog learns to follow you and throws off the excess energy. You got a high energy dog that requires a lot of physical exercise, but not just running and jumping, first is 'The Walk' as prescribed by the Dog Whisperer. Best for you is to get professional help. Maybe your local shelter can support you in finding proper training, so that your dog does not become a casualty. Please act sooner then later, this post of yours sounds very alarming, keep us posted.
Jack Russells are feisty dogs. I know. I used to have one. They are notorious for being quite stubborn, and hard to train.
I agree with your attempts to give him healthier dog food. Some of those food dyes are downright nasty, and have no place in dog food. When you try him with a new brand, how long will he refuse it -starve himself? I know he is used to the old one, and could even be slightly addicted to some of the ingredients (?) but his stubbornness I am sure, will not override his survival instincts. Jack Russells are great survivors!
How long do you let him go without food, when he refuses it? The reason I ask is a dog can go a couple of days quite happily without food. In the wild, predators like dogs and wolves don't necessarily eat every day, twice a day or so like domestic pets. As long as he is drinking plenty of water, he'll be ok. I happen to think that when he gets REALLY hungry, he will relent, and eat....?
I know it seems cruel, but you could try not being the one to give up first on this. Certainly you could give it a try. If, after two days he has still not eaten anything....then you might have to relent, and try another idea.
As for biting, and the other behaviour, he is behaving as though he were in charge, not you. You will have to be very firm, to show him you are the one in charge, and to gain his respect. Cesar Millan has some good ideas in this direction, and they work. Although he has been criticised for some of his methods, he is right about many aspects of dog behaviour.
A Jack Russell is a very energetic dog and needs a terrific amount of exercise daily. One walk round the block on a leash won't be enough for him.
Unfortunately during his rehabilitation, the first thing that is going to have to go is him sleeping with you. He will have to sleep on his own bed, to show him his place in your family.
Dogs without issues can very often sleep on their owners' beds without problem. But in his case it isn't working, and a little separation in the heirarchy between you and him is going to help to show him you are in charge, not him.
You might want to get some Cesar Millan tapes, and see his approach to biting dogs. He believes the owner must be trained more than the dog, and his approach seems to really work. He also believes in heavy exercise to tire the dog out.