My response is regarding allergies in general at specific restaurants and not specific to the allergies you listed since the last time we ate at those restaurants we weren't aware that we needed to deal with those ones yet.
We went once to Beaches & Cream. Lets just say it's a good thing there's visually a difference between the soy and regular vanilla ice cream or else DD14 would have been throwing up our whole flight home (we went mid-day before catching DME to the airport). Luckily it was only me who had a lick of the ice cream before I realized it was the wrong stuff. I just needed a nap in the resort lobby while waiting for DME and coped with the cramps (thank goodness it was only a lick). The CM actually lied to me about having a dedicated allergy milkshake machine (I saw the manager go to clean it after so I asked and she said they just clean it for people with allergies and that it's not dedicated). I will never set foot in there again. Some things I'll give a second chance. Not B&C. I consider it dangerous.
I've read really good things about Teppan Edo with allergies. Because a chef at Tokyo Dining told me that he just can't recommend eating there with our particular list of allergies, I figure Teppan Edo's also off the list. I think it was the sesame that had him worried. With all the reports I've read and the fact that a chef at Tokyo Dining actually understood enough about allergies and cross contamination to be concerned for my daughter's safety, I would not hesitate to recommend the Japanese restaurants at Epcot.