Crossing cold air return ducts with romex
You are correct.  You cannot run NM cable through an air duct.  EMT is approved to run through a plenum, but I can't speak to the specific method you described. Someone with more experience may edit this answer.

The way they ran the NM looks fine.  Some of the holes they drilled might be a little too close to the bottom of the joist, but no big deal.  The wood does look a little messy.  Looks like something I would have done to keep my wife from hanging clothes from the cables. So long as the radius of the bends in the cable is not too severe and there is no friction wear, no problem.