What is this line found above the door on many aircraft?
These are rain gutters.

They are designed to catch rain that runs off the upper surface of the aircraft fuselage and channel it away from the open aircraft door so that the water does not enter the cabin.

This is a close up of the rain gutter over the main cabin door of a Beechcraft King Air B200, showing the channel that would catch and redirect water flow to the side of the door opening:



Source: own work

A jet-bridge may partially deflect rainwater in some cases, but the gutter helps mitigate water exposure for the aircraft interior and passengers. Even gutters cannot completely eliminate such exposure since their size is necessarily limited by aerodynamic constraints. Many operations do not have the luxury of jet-bridges, making the rain gutter especially helpful in reducing the amount of water that might otherwise enter the cabin and pose a hazard to the aircraft or onboard systems.

"Rain Gutters" is the technical name, at least for Boeing aircraft, such as the B777 depicted in the photos in the question.

Here is a Boeing reference to such gutters on the 787:


  Rain gutter: When customers found that water was not being properly deflected
  over a passenger entry door, they requested a change. Boeing relocated the
  gutter to function more efficiently. The solution means happy customers, who had
  fewer water and maintenance issues, and happier and drier passengers. This
  was a significant enough change to process to require certification work with the FAA.