I don’t have time to read this entire thread, so this may already have been discussed.
“The door hinge” and “the door’s hinge” aren’t necessarily the same. Without the apostrophe, you could be referring to a hinge that may or may not be attached to a door. You can have a box full of door hinges. But a “door’s hinge” refers to a hinge that is associated with a specific door. It may actually be on the door, or at least meant to be on that specific door.
I would never use the phrase “the doors hinge” (unless “hinge” is a verb, e.g. “the doors hinge on the left.”).