I can only comment about the cowhide rug idea as that's my area of expertise. Cowhide are still going very strong as a design trend, they haven't been overdone in my opinion. Also cowhide rugs break all the normal rules of standard rugs because they are an organic shape. So they visually take up more room than they physically occupy. So you don't need a cowhide rug to be as large as a wool rug. They also look better on a 45 degree angle to furniture and it is OK for one or several legs to go underneath furniture as this can anchor a group of furniture. For how to clean a cowhide rug there is some great info on my website plus a video that is really interesting.

Lower right. You don't neccessarily have to have all 4 feet of sofa on the rug, but you should have at least the front 2. You'tt probably want a more durable, washable rug because of it's location at the door. Best length would have to be judged based on door location and some sensible distance from TV -your 10 - 20 cm? Out of curiousity, what is the white box in the upper right of the room? Is it movable? If it is, it looks like your sofa could go on that wall, and a light chair (or 2) where sofa currently is. Then the area rug would be centred on the sofa, the chair legs might not be on it, and rug would be away from the patio door. Is that any better?

Love these rugs from West Elm, they pick up the colours in the room and compliment the brick walls also, a round set of coffee tables would be perfect when friends come over then put back together when not in use, some gorgeous lighter cushions would complete the lovely space.

I wouldn't make any suggestions without seeing pictures , and knowing about the load bearing structures -- anything can be worked around , but if it costs $100k to get rid of the hallway , for example , it won't achieve much extra space or light IMO . One thought though -- did you consider swapping the dining with the family room -- at a quick look you may be able to open that wall at the right of the kitchen easy , by disguising existing load-bearing uprights as part of a breakfast bar , for example ? Just a thought on a potentially cheaper option .