Welcome-- my daughter has also got severe egg and pn/tn allergies that we've known about since she was a baby. She's 12 now.
Preface: we've definitely seen life-threatening reactions from shared lines with egg, so we've historically been EXTREMELY careful-- and some of what I'll say below is based in our particular comfort zone that way. Some people are probably a bit less concerned about some of those things, and that works just fine for them. It actually is seeming more probable with every research study that being OVERLY cautious about exposures may make egg allergy more persistant. So do bear that in mind; we've always taken the stance that we avoid things that cause reactions, or seem likely to result in large obvious exposures.
Boxed convenience mixes?
Bisquick. Better Crocker cake mixes. Read labels on those, but if they seem fine, we use them. I do not use box/canned frostings, I also do not use Duncan Hines or any prepackaged cookies mixes at all. No boxed bread or muffin mixes. No corn muffin mix. Brownie mix-- Betty Crocker again. I just substitute the egg called for in the mix, obviously.
I hear good things from others here bout Cherrybrook mixes, though we've never had a chance to try them ourselves.
Pasta? If this hasn't occurred to you and you've never had problems, my advice is to leave this one alone. Finding pasta that isn't on shared lines with any egg at all is an on-going nightmare, trust me. If you HAVE had problems, you might try Hodgson's Mill whole wheat varieties, de Cecco (imported from Italy) or Safeway store brand "O-organics" pastas. Only the latter two are on truly unshared lines, and both are still made in facilities with egg in them somewhere. Pasta has probably been our biggest struggle over the past decade, because just as soon as we found something that worked, the manufacturer would change production and it wouldn't be safe enough anymore. Learned that the hard way a few times, because unfortunately cross-contamination from egg is NOT required on labels in the US.
We've used Keebler pre-packaged crackers, snacks, and cookies. They label very well for pn/tn and 'well' for egg, IME. Important to remember, though-- every label, every time.
Other things that just plain make life better with egg and nut allergy:
Vegenaise. It's a mayo substitute-- and one of a handful that is TRULY like the real thing enough to bother with.
Follow-Your-Heart salad dressings; these are the only egg-free salad dressings that I know of. Everything else is on shared lines. Though I will add the same caution here that I did above with the pasta-- namely, that if what you've been using has been fine... I wouldn't go looking for trouble by calling about it. You won't like what you learn. ALL other commercial salad dressings we've found have been on shared lines with egg-containing ones, and my daughter could never tolerate any of them.
Stretch Island fruit snacks (like fruit leather) are a big staple around here. Your child is a bit on the young side for those still (they are pretty chewy).
Regular Rice Krispie Treats are a pretty good prepackaged treat that is safe for both egg and nut allergy.
Restaurants?
Um... well, mostly? we don't. Sorry. Egg is just too common; we pack food pretty much wherever we go, and we look like Gypsies when we vacation. My daughter eats at one local pizzeria, a local burger chain (pn free, and nearly egg-free), and Taco Bell (selected items; rice and pintos-n-cheese, mostly).