The answer to all your questions is yes. The officer does not have to make themselve "visible". The only real restriction of police officers when performing traffic enforcement duty involves artificial or temporary methods of blocking themselves from being seen. Police can hide behind bushes, signs, trees, or any other object considered to natural or permanent. During hours of darkness, officers may use the cover of darkness as a natural concealment.
Police cannot use temporary means of disguise. Things like covering the car with branches or vegetation, moving temporary objects like signs or vehicles to conceal themselves or pretending to be road or construction workers as a method of traffic enforcement. And most jurisdiction require the police cars to have permantly mounted emergency lighting. There are exceptions to this rule, it varies state to state.
As far as pulling someone over outside of their jurisdiction, in most cases the infraction should have happened within the jurisdiction. But in cases where the public welfare is at stake, any police officer can attempt to make a traffic stop. Also, many police agencies have agreements for overlapping jurisidiction. This is done to provide seamless traffic enforcement and to insure public safety.