An immovable object will not move, no matter what happens to it. If you want to move it in game (such as with an editor) you have to make it moveable, do the translation then make it immovable again. I think making it immovable is more or less the same as not having a body. You should make sure you add all immovable objects to each other's no-collide lists (and your heightmap's no-collide list and vice versa), as I believe they'll still do collision testing on each other otherwise.
Bodies not active will continue to collide with other objects but won't respond to physics themselves. A good example would be a wheel that hangs in the air. They can be moved around by other objects moving however.
A body represents the physical response, center of mass and inertia of a physics object. Since your heightfield is unlikely to want to move much, it doesn't need a body. Just a collisionskin.