Do ETOPS rated airplanes ALWAYS fly with ETOPS requirements?
The short answer is No (as per the regulations)  

The ETOPS regulations state that an aircraft conducting ETOPS operations must be compliant to ETOPS regulations regarding its configuration (based on engine count). But aircraft capable of operating ETOPS flights only need to comply with the Advisory Circular when actually flying ETOPS. In other words the ETOPS AC only applies when the planed route falls under the guidance of the AC. 


  All two-engine airplanes and three- and four-engine passenger-carrying
  airplanes operated under part 121 are required to comply with §
  121.161. This regulation imposes special requirements for ETOPS for these airplanes. These operations are defined as:
  
  (1) Two-Engine Airplanes. These are flights whose planned routing
  contains a point farther than 60 minutes flying time from an adequate
  airport at an approved one-engine inoperative cruise speed under
  standard conditions in still air.
  
  (2) Passenger-Carrying Airplanes with More Than Two Engines. These are
  flights whose planned routing contains a point farther than 180
  minutes flying time from an adequate airport at an approved one-engine
  inoperative cruise speed under standard conditions in still air.




However, how an airline choses to route and use its fleet is up to them. Since ETOPS airframes may require different maintenance or other related things it may not be effective for a given airline to use ETOPS maintained aircraft on Non-ETOPS routes.