The most kid and pet friendly couch material is leather. So far our blue leather couch has coped with 18 years of toddlers falling asleep and weeing on it, cat vomit, Easter eggs melting between the cushions, spilt milk, spilt orange juice, spilt red wine, spilt coffee, and muddy dog paws. It just wipes clean. Our white leather chair is just as durable and is now 24 years old and just starting to look grubby. So go with beige or cream if you want. It would be a lovely choice, just make sure it is leather. Black or mid or dark brown would also look lovely with that beautiful rug but I think lighter colours would look fresher. Don't be at all scared to use light coloured leather.

Agree with above comments - large rug, darker colours to balance the beige. And I also think the art work is too small and too high. However if you are attached to these pieces they could be hung together on one wall, lower, maybe re-framed in similar frames to give them cohesion (black to match your new cushions?). This would create the effect of a larger art piece. (The centre of your art should be at average eye height - and in that room you are sitting so eye height is lower!)

Are wool rugs better than synthetic rugs?

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