Does it violate any regulations for a bag to sit on the flight deck floor?
There are some relevant links in the Cabin Saftey Index document under "crew baggage" indicating that all bags must be restrained. The document points to 


  §121.576   Retention of items of mass in passenger and crew
  compartments. 
  
  The certificate holder must provide and use means to
  prevent each item of galley equipment and each serving cart, when not
  in use, and each item of crew baggage, which is carried in a passenger
  or crew compartment from becoming a hazard by shifting under the
  appropriate load factors corresponding to the emergency landing
  conditions under which the airplane was type certificated.


The question remains if the cockpit is in anyway different than a "crew compartment" but I would say that could be applied to your question. 

However if the bag in question is a flight kit then different rules apply which can be found here:


  A. Flightcrew Flight-Kits. Particular attention should be given to
  compliance with part 121 regarding restraints for any baggage carried
  on the flight deck. Flightcrew flight-kits are not items of crew
  baggage. This policy also applies to aviation safety inspectors (ASI)
  and additional flightcrew members. While it is logical that
  flight-kits be placed so that movement is restricted, the FAA does not
  intend that they be restrained in a manner that would interfere with
  the needs and functions of the flightcrew.


I can not find a further definition of "flight-kit" and/or if you can or cant put your lunch in it.