It is definitely a little difficult to arrange the perfect seating for the space. Although I love all the windows in the room, it brings all these wonderful natural light into the space. I would place the sofa in the center of the room and the arm chairs if you have two one of each side and either a round coffee table (If you do the coffee table I would do it with a glass top so the space looks lighter) or round ottoman. You can mount the TV on the corner wall in a movable bracket.

I have lived in 2 log homes. I understand your problem! Do you have any interior stonework also? Last time I was in this position, I tried to pull the base color of the rug from the lightest tone in the stone on my fireplace, so that I didn't end up with too much going on. Jewel tones do seem to work well, also. I like the last 3 you posted in the comments. It is really hard to pick unless you can see the rugs in person, too. I ended up ordering my last rug through a local furniture store, where I could see and feel samples. They ended up being very competitive on price. I tried ordering online first, and the colors always seemed a little "off" once I got them rolled out in my wood-everywhere house.

https://www.wayfair.com/rugs/pdp/bungalow-rose-allendale-brown-area-rug-bngl8197.html https://www.wayfair.com/rugs/pdp/charlton-home-kaiden-distressed-vintage-teal-area-rug-chrh7678.html https://www.wayfair.com/rugs/pdp/three-posts-centerton-teal-area-rug-w001671539.html look good with art and furniture

For a 14x13, I think an 8x10 would be fine. You might be able to go smaller and have it centered with your furniture bordering it and not on it. I would lean toward a smaller one since you'll be installing brand new hardwood. I think 9x12 would overwhelm the room and hide your new hardwood. https://www.wayfair.com/sca/ideas-and-advice/guides/how-to-choose-the-right-rug-size-T456 https://www.rugandhome.com/blogs/decorating-guide/17369272-rugs-101-selecting-a-rug-size