"fell down at the door" is perfectly correct. It sounds a bit formal. "fell down by the door" is also correct, and is more colloquial. However, those two phrases don't automatically mean "In front of" the door. The person could have fallen down to to side of the door, or, from the point of view of the speaker, behind the door.

I would say I fell in front of the door by saying you fell (have fallen) implies you 'hit' the ground.