Oh goodness, that rash. It does look at bit like my son's eczema, although his is patchy and hers is very widespread. Is it itchy?
I have two son's with cmpa: one diagnosed at about age 3 following years of symptoms (and many other allergies too) and one diagnosed as a small baby.
Yes, you need to remove all dairy, so not just the obvious stuff like milk and cheese, but you need to look at all labels and find any hidden dairy.
What milk has the doctor advised to use instead? You can find many with added calcium, and you could give her a calcium supplement too. I think I would avoid rice milk because it is not supposed to be used as a replacement milk before age 5 (although is okay to use in smaller quantities, just not as a main milk). Soya milk is great, but you should know that quite a large percentage of children with cmpa are also allergic to soya (one of my sons included), so if she does not settle without the dairy, it may be wise to cut out soya for a while too (again, it is a common hidden ingredient). There is also oat milk.
My older son who was diagnosed at 3 was a very unhappy baby, would never settle, lots of runny nappies, tummy ache, and eczema, and also developed allergies to other foods. So quite similar I think. My younger son had bad reflux, lots of wind, no normal nappies, and weight loss. Both struggled to feed and weaned themselves early.
Hope that information helps. Has the doctor has given you some treatment for that eczema? Hopefully it is cmpa and then things will improve for her.