Clarification on flight hours as compensation
There are a few different questions in here, some are FAA regulated and some are a matter of the terms of your employment


  If I have a CPL and fly for hire, paying no expenses of the flight,
  are my flight hours considered compensation in addition to my wage?


Maybe, its somewhat up to your employer, but more importantly its not really what the FAA cares about. You have a CPL and are presumably flying for a legal carrier/operation. In the eyes of the FAA they don't much care how you are paid beyond that. Practically the legislation is more to ensure that people who are getting paid are properly licensed and working for a proper operator.


  Do I have a legal right to those hours?


As Juan notes if you were PIC (or SIC) properly certified in the aircraft then the time is yours. 


  Is a company allowed to, as a condition of employment, require that I
  don't log hours while flying their aircraft?


That seems like it would be encouraging you to not follow a FAR which is not a great idea. 


  If a company suddenly and without reason sticks me behind a desk
  instead of having me piloting an aircraft, would the loss of flight
  hours be considered a pay cut?


This depends two things, your existing contract with your employer, and if that contract changes when your job changes. Often times with a role change your contract, or the terms of the original undersigned contract can be changed. If you were a pilot with a Minimum Monthly Guarantee in your contract you may be entitled to keeping your pay-rate. However if the new job comes with a contract change or shifts where you are under your current contract you may not be entitled to anything.