I would reconsider it being the “new” dog. Although your cat grew up with the other dogs, and their smells are part of who she is and what she’s familiar with and has a history of no need to mark territory with them ….. that is not the case for a new animal added to “their” territory.
I can’t imagine it’s all that cool or possibly sanitary for the dogs to be eating near a litter box.
It certainly doesn’t cost anything to “remove” the dogs to another feeding area to see if the cat reacts differently.
As to getting the smell out of carpets. We’ve had great success with Pet Force. It’s had a long history of good experiences and the cost is so totally cheap! Here’s a site: http://www.healthypetcorner.com/shop/Pet_Force__32_oz_p_16226.html