Bonnie, I have an open floor plan from the LR to the DR. They are right across from each other and only separated by a walking path to the FR. I live in a 2300 sq. ft townhouse. I have wood floors going through 90% of my home. I don't follow many rules of thumbs when decorating but for this one: with solid wood tables use a solid rug. With a glass table always use a patterned rug. You don't have to have a rug in every room. Here is the reason: A glass table can be seen thru and you can actually view the beauty of a patterned rug at any angle. You can't do that with a wood table. Besides, wood tables give a warm feeling on their own. My feelings for eating areas is to use no rug since crumbs (people have the tendency to step on them once they drop) and spills do happen. But that's me. Many on this forum would disagree.

I also like the last one you posted and think it will work well. I also have two rugs close by in an open floorplan. What I've found works well is to use similar colors, but to have the rugs vary either in pattern or in the predominate background color. One rug has a gold background with rust, brown and sage accents with a black border; the adjacent rug has a sage background with the rust, gold, and brown accents. They need to be different enough to stand alone, but have some elements in common in terms of style and colors to link them together.

Is the bench in front of the fireplace small enough to fit on either side of the fireplace under the window? If so, put it there and get a matching one so you have benches on both sides. If not, get two benches that fit. That lounge chair/ottoman combo to the right of the FP feels out of place and isn't part of the conversation area. I would find another home for it. This arrangement has the added benefit of not blocking one's view of the FP from across the room. I agree with continuing the jute rug and removing the hide rug from that area. I would add some throw pillows for the white chairs that have more of your sofa's chartreuse color in order to tie the two spaces together. If the bench is moving, you now have room for a small, low coffee table in front of the chairs. Lastly, the accent wall color you've chosen is lovely, but doesn't really coordinate with the wallpaper. I would have picked up one of the lighter greens from the wallpaper to pull it all together.

I love the rugs JudyG posted. The tones work well together, cohesive but not matchy matchy contrived.