The cable in these pics is a form of Romex that has not, I think, been made for decades. A flat three wire cable, with an outer jacket of soft, grey PVC. The hot and neutral wires were insulated (TW, I think), while the center ground was simply wrapped in paper. The insulated conductors were stranded. The ground was solid, but a gage smaller than the insulated wires.
I'm pretty sure that no edition of the code endorsed running the cable across, and through, the doorway. This is 'as installed;' it did not simply fall free. I also expect that even a handyman could have run the cable in a neater manner across the wall!