What's the purpose of the chair? If it's not for use and just decorative do not put anything. Leave it blank and allow the space to breathe. So many people feel compelled to fill the space because it's uncomfortable much like being bored or not talking. A beautiful piece of artwork would do well at the top of that landing.

Furniture arrangements should have a purpose. Having someone sit in a chair way over by the stairs would be like putting them in time out. You need a rug for your seating area, and you probably have enough room for a pair of chairs on the other side of the coffee table.

Yes to a chair to the right of the fireplace. The space is empty and needs "something". A chair in this position -- angled towards the other seating -- will create a bit of a circular seating area /will look good and be functional. The chair will need a small table and lighting -- perhaps a swag lamp overhead? Don't go with too big a chair, or it could become a traffic obstacle.

What I don't like in this room: the fireplace stone is so lovely, but it's ruined by that big TV.

I wouldn't put a chair there, that would seem a little awkwardly out of place. But I agree the wall does need something. I'd probably go with matching cabinets on either side of the fireplace. Or maybe a tall plant for a little bit of greenery. 

I do think  the room is crying out for a rug. A rug would  take less emphasis off of that bare wall because your eye would go to the rug and break up the bare floor to the bare wall. 

I'm not bothered by  the TV above the fireplace. The room is to live in not for a magazine picture. People do watch TV