Hi and welcome!
I'm sorry your DD is having food reactions - it's so hard when they're little and can't really tell you what they're feeling!
I'll try to address everything you've brought up in your post, but I'm sure others will chime in, too.
1. Food allergies should not be diagnosed by testing alone because there is a high rate of false positives. Also, back when my kids were little, our doctors felt like allergy testing was even more unreliable in very young children. If your DD has eczema, that can also affect the results.
2. The allergenic proteins in milk and eggs can be affected by heat, so some allergic individuals are able to tolerate baked milk and eggs - the longer the heat exposure the more those proteins are altered. This is NOT true for peanut protein.
3. If your DD can tolerate cheese, then I would question whether the milk allergy was a false positive, though.
4. Most ice creams are cross contaminated with peanuts, so since your DD is peanut allergic you will need to learn about cross contamination and avoid those exposures.
5. As to why your DD did not react the first time she had peanut - the first exposure to an allergen often will not cause a reaction because the person is not sensitized. Only after a person has been exposed can she become sensitized and begin to react. Also, people can develop allergies at any time - even after decades of not being allergic to a food.
6. If your DD eats something and has skin symptoms in places the food did not touch, then that is a systemic reaction and could be very serious. If her skin is swelling, so could her throat or lungs and she's not old enough to tell you what's going on inside. I would strongly urge you to see a pediatric allergist and get an action plan for recognizing and treating reactions (and also to help you figure out what her true allergies are).
7. Many children outgrow milk and egg allergies, but I think the number who outgrow peanut allergy is lower. A pediatric allergist will be able to guide you as to the likelyhood of your DD outgrowing some or all of her allergies based on test numbers, reaction history, etc.
There is a steep learning curve when first having to deal with food allergies, but it gets better!