My daughter (22months) is also allergic to milk and is a very picky eater. I would recommend going to a store like Whole Foods. Make sure to look at labels...you would be surprised how many things have milk in them...I'm sure it is the same for eggs. Look for the brand Ians. They have all sorts of foods (chicken nuggets, fish sticks, etc.) and they are free of all allergens...my daughter particularly likes their alphabet french fries. The french toast were a little hard, but your child may like them. At whole foods they also have two fresh vegan cookies as well as other boxed cookies in the aisles that are allergen free. Also at Whole foods, they do have a mac and cheese in a box that is milk free (you would have to check about the eggs). My daughter hated it, but I have heard of other kids that like it...it might be worth a try.
Some other things my daughter will eat are Applegate hot dogs (chicken, turkey or beef), Bell and Evans chicken tenders, certain types of Van waffles or original waffles that have cookie monster on the front. I also buy very firm tofu, cut it up into bite size pieces, bread it with panco (Japanese bread crumb) and put it in a pan with a little oil. My daughter really seems to like it. Sometimes I also use smooth tofu and mix it into her pasta sauce over noodles. My daughter will not drink soy or rice milk. I do use rice milk when I make her things that require milk. She eats Silk soy yogurt and there are popsicles and ice cream that are made by Soy Dream that taste pretty good.
Just a note of caution with cheese...really read the labels. I have yet to find a cheese that does not have some sort of milk product in it.
I hope some of these ideas help you. It has been a very challenging journey for me...I wish you luck.