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.

Hi, did you mean to attach a photo?

Look for a piece of carpet and get it edged. Ask at carpet shops to see remnants or off a new roll

I think it's fine without a rug. I think rugs are good when it's a large open area and it helps you define spaces. This dining area is well defined without needing a rug