I can relate to both of those symptoms. They don't happen to my DD for baked egg, only direct egg.
Even right now, DD does not want to be around when eggs are cooked. It's not an "I don't like the smell reaction, I believe it is a flight to safety reaction" I air out the house if I cook any for the boys.
Ditto for my DD with egg exposure and eczema....I could not believe what gorgeous skin DD had when egg left her diet!
I hope there is no other progression of symptoms for your DD.