You never said anything about a static body being moved with the mouse. This is not really something that can happen in the real world, and that is why we don't try to simulate it in the engine. There are several problems you have run into related to the
1. Static bodies can by definition not move (they are static...) and thus, they can't apply force to sleeping bodies.
2. Static bodies can't create contacts with other static bodies. They might overlap, but since none of them can move, there is no need to create a contact between them.
The only way of doing this reliably in the engine, is to use joints to move the body. What are you trying to achieve?