What is the FAA definition of "takeoff" and "landing"?
According to the FAA FAA Airplane Flying Handbook, there are three components that make a takeoff:


  
  Takeoff roll (ground roll) is the portion of the takeoff
  procedure during which the airplane is accelerated
  from a standstill to an airspeed that provides sufficient
  lift for it to become airborne. 
  Lift-off is when the wings are lifting the weight of the
  airplane off the surface. In most airplanes, this is the
  result of the pilot rotating the nose up to increase the
  angle of attack (AOA).
  The initial climb begins when the airplane leaves the
  surface and a climb pitch attitude has been established.
  Normally, it is considered complete when the airplane
  has reached a safe maneuvering altitude or an en route
  climb has been established.
  


So to have a take-off you must have these three components.  That would include a touch and go as well takeoffs from a full stop.

As far as an official definition of landings I didn't see anything official in the FAA documentation but I would say that the airplane touching the ground would count as a landing.