My son does not have the same issue (the eggs) but he is allergic to Red Dye. And that was a big issue before school this year. (Pre-K)
In your situation I would just ask for things to be substituted, not change their menu but just make substitutions for your child. That doesn't seem like to much to ask for. And that would probably take away any lunch issues. As for the parties, could u tell the teacher u have a non-egg cake recipe and you'd be glad to make a small cake or even some cup cakes, for the parties if u were given enough notice? If that is not possible, (to make something from home and bring it to school) then what about store bought cookies, or frosted animal crackers, or maybe even caramel apples. I'm not sure what all has eggs and what doesn't, so I am kinda coming up with foods off the top of my head. But maybe next time u go to the store u could check the labels of some different things and get ideas that way.
But we did the same thing with the meetings and all, and we got some of the strangest questions, the funniest was when I was asked by someone at the school if my son could have ketchup? And I was like ya, why? and her response was well it's red so I just thought he couldn't have it. So I went back over his information again to clarify anything I might have left open for question and (knock on wood) there hasn't been a problem since. But for the first two weeks of school they wouldn't let him have anything that was the color red, no matter if had red dye or not. I try to laugh at it now, but it was a little stressful, but hopefully this will not be an added stress on you. The schools are usually more than glad to help in any way possible. So that is a good thing.
Hope I was able to help, if not, sometimes just knowing your not alone is help enough.