I totally am with Michelle on this. My son also has severe food allergies, dairy, wheat, nuts and eggs. I breastfed him for 2 years. You need to take this seriously. Some people don't get the seriousness of this issue and dismiss it as an intolerance. You have evidence that this is not the case. My advice is first of all to ALWAYS carry benadryl with you everywhere. And get an Epipen prescribed ASAP if you don't already have one.
My son was also diagnosed at 6 months old and at nearly 3 still has all his allergies. He also used to break out randomly, just like your daughter, I know how frustrating it can be. That's we we always have Benadryl. they make the single dose now, that is premeasured and can be thrown in your purse, diaper bag, glove box, etc.
You can buy an egg substitue at Sprouts or Central Mkt. It's in the baking aisle in a box. It's a powder that you mix with warm water. It really only works as a binder for baking. I already never ate eggs, so that was an easy elimination for me. I gave up baked goods while nursing too. All cakes, cookies, etc. will have eggs in them. It was hard at first, but I got used to it. Plus, Sprouts has plenty of egg free treats. Also, the flu vaccine has eggs in it, so make sure she doesn't get that vaccine.
You will need to read ALL labels from this point forward and learn the different words used to describe eggs and milk. Albumin (eggs) is one I remember off the top of my head. Most labels will tell you at the end of the ingredient list what allergens are in the product.
My son drinks non-GMO soy milk and eats Silk soy yogurt. My pediatric allergist told me that goats milk and cows have a high crossover rate, Meaning if she's allergic to cow's milk, then it's highly likely that she'll be allergic to goat's as well. Butter and margarine both have dairy too. We use Earth Balance now.
Sorry if I've overwhelmed you, it's just that I remember how scared and lost I felt when I first got the diagnosis and want to help you. I have an allergist I love in Richardson if you want the #.
Good luck to you.