Dubai is a 3rd world police state and they do know your every move. Do they act on it? Depends. If they want to get you they act on their information but generally if you play by the rules and keep your head below the radar you will not get bothered.
There is always that threat though so do be careful.
They can go through your apartment ANYTIME they want to as again it is a police state and you have no rights. Generally they do not as long as you are not dealing drugs or talking to al Quada.
Emirates is basically the same with the lack of rights. They do whatever they want to.