Yeah, I highly doubt that exposed romex like that will fly. You could put it in conduit for the short run. It is Ok to do that for protection purposes even though you are otherwise not allowed to put NM cable in a conduit. Just shoot a piece of conduit straight up and put the rest of the wire in the joists above (not on the drywall). That should pass.
Your dryer box is grounded improperly in your video (I watched it today while waiting for coats of finish to dry on my latest door parts). The box itself needs to be directly grounded with a ground screw since it is conductive. All modern boxes have at least one pre-threaded ground screw hole in them so that is easy. A lot of times I just cut the cable extra long and wrap the ground wire around the screw and then run it to the outlet. A very picky inspector might complain that it doesn't quite go around the screw far enough (maybe...a judgement call), but I find it does the job. Then I cut the insulated conductors shorter as needed as you only need the ground wire to be longer. You can also just pigtail a ground wire from the box to the incoming ground and another pigtail to the fixture.
Do keep in mind that you will need to ground all metal boxes like this, so don't get too far ahead of yourself on that and forget. The metal BX sheath does not count as the ground either, but it must also be grounded - which it will automatically be via the box ground
