I would eliminate all forms of dairy to start. Your daughter may be very sensitive or she may not be. If you eliminate *everything* to start, then you can start to trial things back in once her skin clears up.
For my dairy intolerant 18-month-old nursling, I have to be very strict on dairy. He actually reacted (GI upset, colicky stuff) to lactose in a cold tablet! So yes, I'd pull it all to start. You can find great info & avoid lists here: www.godairyfree.org, www.foodallergy.org, www.nomilk.com (this one might be a little political, IIRC), http://kidshealth.org/parent/misc/milkallergy_cutout.html.
Places you wouldn't expect to find dairy but might: pickles, lunch meat, deli meat (shared cutting surfaces with cheese as well as lactic acid as an ingredient), olives, rice cheese, items labeled 'non-dairy.'
As for butter, Earth Balance has a couple that should work for you... I think the light and the new olive oil ones would both work. And I'm pretty sure some of the Sara Lee breads are milk-free.
You can try soy but it's also fairly highly allergenic (one of the top 8, along with dairy, egg, wheat, fish, shellfish, peanuts, tree nuts) so I'd try not to add too much of it in until you know if she also reacts to that. (Soy and dairy intolerance/allergy often go hand-in-hand.)
I'd be wary of goat's milk, as most who react to cow's milk also react to goat's. I'd also be wary of nut milks, but I do have a nut allergic child, so I'm always suspicious of nuts. ;) Rice milk would be a better bet, statistically speaking, as would oat milk, hemp milk or potato milk.
Forget about cheese, IMO. Most cheese subs are semi-horrible at best. I sub in guacamole (but if you don't make it yourself, ask! Lots of restaurants put dairy stuff in it!), hummus, etc.
You can do this! It is hard, and you might want to find a support group (I'm in a great allergy group on Yahoo... it's called TerrificKidsWFA.) to help you get through the tough days. I found dairy to not be as hard to give up as I thought it'd be (I'm a huge cheese eater), so maybe that'll be the case for you.
Best of luck! Email me if you have any other questions or need support!