Right, Kathy -- You start with the obvious: basic needs such as food, need for sleep and so on. Also, the 7 year old has to be dealt with. Part of the turmoil (perhaps the largest part) is the lack of any basic discipline previously, and the feeling of being jostled around in foster care. Anger issues can be present at such a young age. The idea of rewarding positive behavior and punishment for improper behavior might reinforce his will to do good. Does he seem starved for affection? This might be difficult, given the number of children in the family.
One school of thought is if a child his age is acting up to just let him continue until he realizes it isn't doing him any good. Perhaps being removed from the scene of the crime can help until he calms down. He might be old enough to understand being told: "When you can behave, you can join the others." It's telling him he has the responsibility to shape up.
As you also pointed out, he needs some physical activity to burn off that excess energy.
There may be something troubling him (he's facing a lot of problems for his age), and perhaps a therpist's evaluation might be helpful. There's such a combination of factors that can influence his behavior, it's difficult to pinpoint. Determining what the problem is (or at least the major problem) is a key to solving it.
I have tried the whole ignoring thing while he is screaming and having a fit in time out (keeping an eye on him for safety, but for the most part leaving him alone); however, at that point his parents hear him screaming and come up and take him away to talk to him, which I think might be counter-intuitive. If he knows that screaming will get him alone time with his parents then what's to stop him from doing it?
I just don't want to be the young nurse with no children who's trying to tell them what I'm trying to do to discipline their child. I guess it's a Catch-22. It makes sense to me though that eventually if he screams and kicks and throws things but nobody comes to his rescue, maybe he'll realize it's useless.