What is this protrusion on the door?
It contains the evacuation slide. From airspacemag:


  An escape slide sits inside a carbon fiber pressure cap covered by a casing of material similar to the aircraft interior walls—that big square box at the bottom of an airliner’s interior door.


Here's an image from Boeing:



Image from Boeing.com

According to Boeing, the positioning of evacuation slides inside the doors reduced the deployment time.


  By 1963, improvements in materials and inflation systems reduced the weight and bulk of the slide system, making it practical to move the slides out of the ceiling to the lower inboard face of the cabin doors (see fig. A). This location resulted in a still more efficient escape system, reducing the time needed to ready an escape slide from between 20 to 50 seconds to between 18 and 24 seconds, including door opening time.