@jean, to close it off? If you do it yourself its cheap as hell. Just rip all the white trim off. Screw in 2x4's every 16 inches. Cut a piece of drywall to the size of the hole (you'd have to measure the thickness, but its probably 1/2 inch or 5/8). Screw it on. You might get away with just patching it but taping it would be better. Sand, primer, and paint, done. If you wanna do a more thorough job. Then you cut out the drywall from the floor to ceiling at the studs. Tape, patch, sand, prime, paint. Its really not that hard. Like a half days work. Most it would cost is around $30 not including primer and paint.

Your rug works with the couch, and it also works with the color of the dining room. The dining room color would work very well in your living room. It is the beige that is throwing everything off. Its the living room that needs painting, not the dining room. In the dining room, I would paint that white table and chairs black, or a much darker color. Then add color to both rooms for accent colors in pillows, etc. Black coffee and end tables would add a lot to the living room too. It may be easier to paint the dining room, but, it wont solve your problems in the living room. Its still beige, and it still doesnt work. I love your couch, and your rug. Good choices.

Not understanding the "half wall" comment -- hard to give advice without knowing what the rest of the layout looks like. Right now the wall and doorways control the flow of traffic in a certain way. If/when you remove the wall, what will traffic patterns be like? Let's assume you can successfully relocate the heating and solve the flooring issue...appears that you have pretty large rooms, and will have the opportunity to arrange and rearrange until you find a furniture layout that works for the space. You might consider finding an on-line app or use a furniture store's app that allows you to put in a room's dimension and then put in furniture and move it around -- it's kind of fun.

Removing a wall in certain types of layouts is not always the best idea because it creates a bigger but undifferentiated space. Say this were a wall without the bookshelves. Right now you have two of your largest pieces of furniture essentially 5 inches apart, a big hutch and a sofa. If there were no wall there you would never have two separate-function furniture groupings 5" apart, or back to back like this. You've lost a natural spot for both the sofa and the hutch. You have another spot for the hutch, but you would probably not float the sofa too close to the dining area, so where does it go? Sometimes opened up spaces just end up looking like open plan furniture showrooms, with groupings one against the other and while it's more open, I am not sure a lot of function is gained. Just things to consider.