Can an aircraft get certified for different runway lengths through the FAA based on the amount of payload it is carrying during said takeoff?
Generally an aircraft will be certified for a range of runway distances based on headwind/air temp/weight, a correction chart will be provided in the POH allowing you to compute the runway distance for various conditions. For example here is the chart for the PA-28-161 I trained on



The big planes have similar charts, you can find the 737-800 numbers here and if you want to go to the really heavy metal the 747 info can be found here.

The AC that covers aircraft certification can be found here and interestingly the distances are computed from standstill to the point at which the lowest part of the aircraft is 35ft above the ground. 


  (1) The takeoff distance on a dry runway is the greater of the two
  distances depicted in (a) or (b) below (see page 63 of above linked doc). The distances indicated below
  are measured horizontally from the main landing gears at initial brake
  release to that same point on the airplane when the lowest part of the
  departing airplane is 35 ft. above the surface of the runway.


Interestingly the FAR does not seem to address the weight at which the takeoff distance tests are done.