My 12x12 register hole is in the floor or ceiling depending on which floor you
Air flows either way dependant upon the temps in the rooms above. Right now its piling up dog hair in the ceiling below meaning cold air is coming down and must be moving warm air toward the stairs.
I run my furnace blowers to cool the house and move air around when the stoves get too hot. The funny burn smell when your furnace kicks on tells me you
have an issue with the filter in your plenum. I use micro-allergen type because they capture coal ash dust and dog dander that would settle in the rooms and the plenum which would give the odor you
describe when your furnace kicks on.