You've just proven why I say that parents or guardians are the best detectives in determining their children's particular allergens/irritants. You did an excellent job in articulating your observations. The good news is that I strongly believe that you've positively identified your daughter's allergens/irritants. The bad news is that I believe that your dog (in particular, the pet dander) is a major contributing factor to the persistent eczema flareup.
Animal dander is dead skin that is continually shed. In the cases of animals that serve as household pets, it is referred to as pet dander. Although you've taken the precautionary measures of separating your daughter from the dog, the remnant of the shed skin (pet dander) remains in the air, fabric, upholstery, or carpeting throughout the house. Therefore, even though she's separated from the pet, the pet dander may still be present in her surroundings. Also, if you have a central heating system and even if the dog is kept in a separate room, the pet dander could be circulated in the air through the vents. For people who aren't willing to part with their pets, placing an air purifier system in the effected child's bedroom would be a good alternative. Another precautionary measure is to wash the pet weekly. It may not always be that your child will have an allergic reaction to pets; however because infants and their immune system are at the most vulnerable stage (new to the world, new to the environment), an introduction of a pet at a later point in their lives (as they grow older) may be more appropriate. It's much like the introduction of certain foods or drink; it's a gradual process.
Purchasing the anti-dust mite bedding and mattress covers is a very good option for alleviating your daughter's eczema flareups. Getting rid of the synthetic clothing and changing bath products was a smart move as well. To remedy the overheating problem, it is suggested that infants with eczema wear 100% cotton fabrics and that the clothing and bed linens are layered to adjust to temperature changes. Although it's a lot of work initially, it pays off very quickly and in the longterm when all three factors have been successfully addressed (internal, external, and foreign). My nieces' severe eczema breakouts are distant memories for us; and for them, they don't even remember what the suffering was like. I do believe that this will soon be the case for your daughter as well. We fought so strong in the beginning because my nieces' immune systems weren't able to effectively fight. Because we addressed the immune system with good nutrition (water, fruit, and vegetable) on a consistent basis throughout their growing period, they've developed a healthy immune system; which has taken up the fight where we left off, resulting in less work for us all. I guess the point that I'm trying to make with you is that the fight to heal the skin of eczema flareup will not always be such a frustrating and hopeless struggle. There is no need for you to fear it; but with your gained knowledge, there is a need for it to fear you.