I have wool rugs, and they hold up very well to kids and pets. If you get any kind of pattern, even if it’s light in color, they hide stains. If you don’t mind spending the money for handwoven wool, they are really pricey though, you can keep them forever because they can be cleaned. Otherwise tufted wool works well too. I have a couple tufted rugs that are 10 years old and still have life left. I personally bought a secondhand handwoven wool rug from a very wealthy woman who had had no pets. The rug was in perfect condition...it’s probably at least $2k rug, and i bought it for $150. I had it professionally cleaned when I first bought it, and since then have cleaned it myself every spring with my bissell carpet cleaner. I pick a warm day, powerwash an area of the driveway so it’s clean, and I clean it and then let it dry completely in the sun before bringing it inside. I do the same to my tufted rugs. If you want a trendier rug, that won’t last as long but will be much less expensive, you could just do that as well. There are plenty of options on overstock. An indoor/outdoor rug may work too, but they’re not very soft.

There won't be a square rug in any ready to go rug. However, there are unlimited fantastic broadloom selections that would allow you have something/anything bound to a perfect 12 x 12., leaving you a nice 18 ish inches all around the perimeter. In fact, the best and newest broadlooms have been designed with just that capability in mind. They KNOW everyone wants hardwood, they know they also want rugs, and rooms have gotten larger over the decades. You'll find organic and chunky "green" wools, great patterns, stries, ..............even a sisal can be customized to the inch. A 9 x 12 will be the next best thing in ready made, although it's a size that weirdly gets passed over for 10 x 14 many times. And that is a bit less than perfect, for you.

I have a small apartment which requires rugs (condo rules), and I have a rectangular rug in my living/tv space, circular rug in my desk/office area a couple feet away, then another rectangular rug in my dining space. I like that the circular rug sort of breaks up the two major areas (living and dining). I also have both rectangular art and circular mirrors/hats up on my walls, so I like playing with shapes. Per your photos above, I think that the rectangular and circular rugs would look great! What size rugs are in the pictures above? I hope this helps!

Your space can handle it! If you go for round I suggest a pedestal so no issue with legs. Large square tables are rare and I view them as a more modern or casual look. We have a round that is 54" that has casters for ease of moving / cleaning. 6 chairs around it at all times. 4 use it regularly (we have only ONE dining area in this house) and sit next to one another so it takes some getting used to - imagine talking to someone next to you at a wedding reception. We have a 22" lazy susan which makes it easier to pass a serving dish. If we bring out our folding chairs (those white ones in a banquet hall you can get covers for) we can do 8. We have a 8' round rug that I wish was a little bigger.