The rugs you've shown with your sofa are too small. I prefer the extensions of your sofa to be completely on them with a foot or more of the rug extending past the outside of the sofa. I also like the rug to slip under the front 1/3 of the straight part of the sofa. That said, I think it should be as wide as the wood insert that forms your hearth. If you can find one large enough, an indoor/outdoor rug could be your solution to the beachy look and easy cleaning. You can take them outdoors and hose them off. Here's a 9x10 so they can be found but you'll have to do some shopping online and you may have to settle for a more casual pattern than your FP trim calls for. Do you have a beachy picture to hang above the sofa? Once the kids are old enough to play in their own rooms, I would place the TV right or left of the FP above a distressed credenza that repeats the themes of your FP.

Pulaski Accentrics Home Alexandreah Credenza, Aged Patina · More Info

EmpirePatio Maverick Outdoor Patio Rug, 9' Long x 12' Wide, Royal Blue · More Info

Here are two rug options for your beach/cottage vibe. The images below are from our design projects.

Azalea Cottage - Living Room · More Info

Driftwood and Seaglass · More Info