I'm hoping someone will have some insight on this combo as well. I'm in the market for the 200-400 or the 400mm 2.8 I have the 70-200mm 2.8 and it is a fantastic combo. Even with the Nikon teleconverter tc-20e iii attached it is fast auto focus and sharp.
I do not have the combo. But I can tell you that no camera "balances well" with the 200-400 handheld. You are going to have the lens mounted on a monopod or tripod or you are at least going to handhold the combo by the lens collar. The camera is just along for the ride when it is connected to that monster. If you were to operate the camera by holding the camera with no lens support, your lens mount would bend (over time anyway).
I try to avoid lifting my D3 with the 70-200 mounted on it without holding on the 70-200 lens collar. I certainly would never attempt this with a 200-400 on any camera.
If you have enough light to shoot with fast enough shutter speeds for handheld shooting and you are holding the lens collar, I am sure that the 200-400 will take superb images with the D7000. But I would recommend a monopod.