Selket gave you all the info you need. I just wanted to say that we have been to Disney 3 or 4 times with my peanut allergic son and it is one of the easiest places to deal with. Honestly, I feel that it is the only "safe" place that we can vacation with him. We have a worse time of it when we go to the beach that is 3 hours from where we live.
Just make the ADRs anywhere you like and let them know of the allergies ahead of time.
I guess, though, the worst places for my son were Kona Cafe and 'Ohana (although we ate at both). Both of these places use a lot of nuts in the kitchen (Kona is worse) and he had very limited choices there of food and dessert. I kind of felt nervous at Kona. 'Ohana was "iffy" because of the peanut sauce. That wasn't a big deal because we just ordered chicken tenders from the kitchen and he didn't eat the 'Ohana food at all. Now that he is older though, we just probably won't go there because there wouldn't be much he could be safe with.