Would dropping the picket fence from higher above the photogate change any of the parameters you measured?

Dropping the picket fence from a higher height above the photogate would not change the parameters you measured, assuming the initial velocity of the picket fence remains constant. The time it takes for the picket fence to pass through the photogate and the acceleration due to gravity acting on the picket fence would remain the same. The height from which the picket fence is dropped would only affect the initial velocity of the picket fence, not the parameters being measured.