Don’t know what a round nesting sofa chair is, can you post a photo of what you had in mind? I wouldn’t go with gray flooring, it will clash with your fireplacel. An area rug would look good in this room, also some end table with lamps.

No round nesting chair. They are notoriously hard to work with. Too big and it certainly wouldn't fit with your other furniture.

Even a papasan chair isn't in keeping with your other furniture.

Maybe not a round nesting chair but adding in modern, rug, end tables and a coffee table will look great. I would go with wood flooring for sure.

I can only help you with the colors. In your room, here are warm colors: fireplace (brown and orange), floor (tan), couch (sand) - here are cool colors: wall (white). If you get gray floors, that will clash with the warm colors. I'd go with a warm color for the floor and walls.

In my opinion, I like a room in all cool colors or all warm colors. Here are warm colors: off-white, sand, cream, golden yellow, warm pastels, peach, tan, orange, orange-red, beige, olive green, brown, rust. Here are cool colors: white, bright cool yellow, gray, silver, taupe, black, navy, purple, blue red, royal blue, emerald green, forest green, violet, fuchsia, bright hot pink (yes, this is a cool color!), ice pastels, teal, aqua. If you need more info, there's a book that is out of print, but you can get it on Amazon: Color Me Beautiful, by Carole Jackson.