Whelk - you say that your dd2 hasn't reacted to dairy in your diet, but you also say that she has eczema. This may be her reaction to the dairy that you are consuming. The only way to find out would be to go dairy free for a minimum of 2 weeks (it takes that long to clear from both of your systems fully) and see if her eczema improves.
Re removing eggs from your diet. As I understand it, there is positive proof that cows milk protein passes through in breastmilk, but other foods, whilst sometimes suspected to pass through, haven't been proven to do so. On that basis, and since your dd2 hasn't been shown to have an egg allergy (as yet), I wouldn't cut them out, but would prob avoid an excess of them.
If you are going to be a dairy free household for a while, then other products that you need to know about are:
Butter substitute: Pure sunflower spread, vitalite, M&S sunflower spread. Almost all other marg still contains dairy.
Milk: Oatly (with calcium - not the organic one, which doesn't have any). If you want soya milk for you (I prefer this on my cereal to oatly) then I like the fresh blue alpro.
Cakes & treats: Mrs Crimble range - lots of lovely stuff here, esp coconut macaroons and bakewell slices. Respect organic carrot cake or banana cake (made with oil rather than butter). Dark choc - Divine or Green & Blacks (there's a big saga about a change to their food labelling over the last year or so, but even though it says 'contains milk' it actually doesn't in the dark, the maya gold and a couple of others). You can also have Elizabeth shaw dark choc mint crisps.
Biscuits - asda smart price digestives or waitrose own brand digestives, Waitrose almond biscuits, fruit shortcake, bourbons (surprisingly) are all dairy free, but always label check because recipes do change all too often. It's a bugger finding a biscuit to eat at a toddler group though!
You will need to label check everything I'm afraid though - even some ham contains milk!
Any other specifics you need to know re food alternatives then just ask.
Oh, and I can recommend a good cake recipe book for dairy (and gluten) free cakes too if you like to bake (they do still contain egg, but is there an egg replacer that you can use?) - it's this.