My son is allergic to wheat, soy, egg, dairy, peanut, tree nut, shellfish, sesame, buckwheat and cottonseed. We don't eat out around home but go back to Disney because we can eat there.
Every time you get to a dining location, just ask for the chef or manager. They have allergy menus now and that is an option because all of your allergies are top-8, but you can still ask if you have questions or concerns about cross contact and you absolutely should not feel bad doing so. We find that table service is just easier for us to deal with because they have fewer items that are pre-made. The sit down kids meals now usually include one or two entrees that are major allergen friendly to start with so that will be helpful.
Counter service will take longer... often about as long as a table service meal while at table service he usually gets his allergy food around the same time the rest of us get the normal food and often before we do.
I would take a look at menus and there are certain places, like some of the EPCOT countries where if the menu is really dairy heavy it can be a bit of a hassle... like I wouldn't recommend going to Italy for pizza unless she is a huge salad fan. I've read reports where the table service in Mexico can have a bit of a communication barrier issue with dairy allergies and the French menus have cheese on everything.
Where are you staying?
For breakfast, you are probably looking at allergy waffles and a meat/fruit side every day. A few places in the parks are only going to offer you the van's frozen waffles but at resorts you can ask for the mickey shaped allergy waffles. I would not- absolutely not- plan to have a quick service breakfast in the parks unless you pack her food from home. They are going to look at you like you have three eyes. Everything has wheat, everything has dairy and everything has egg. It's just not meant to be.
If you are on-site and want breakfast to be quick or on the go, I would have an adult run over to your resort quick service and pick up breakfasts and an order of the allergy waffles. My kids will eat those while on the move and they hold together well. Once we figured out our kids loved them I actually would bring a baggie hand each kid a waffle and put the one or two we had left in the bag. When they wanted a snack in an hour or two we didn't have to stop and we knew we had a safe backup snack.
Is it sad that this is one of my favorite parts of Disney, that somebody made my kids fresh waffles every morning? ETA- the ONLY time anybody ever stole something from our stroller at Disney is the time somebody stole my waffles in a baggie! I'm not sure if it was a squirrel or a person but they had good taste.