Not sure if you`ve had to make this cake already. I would suggest going to a specialty store (like a health food type place) and ask for a vegan GF cake mix. It`s probably your best bet because GF scratch recipes tend to be rather complex, and call for a variety of expensive harder to find ingredients. Milk derivatives are a common additive to processed foods and can be under a plethora of names.
Also not sure if maybe I missed something but eggs do not contain lactose or any dairy. I hear that a lot about dairy allergies and lactose intolerance but eggs are 100% ok unless they have an egg allergy.
Becel sells vegan margarine, which is like salted butter. Earth Balance is a very popular brand to look for. I`ve used vegan Becel to replace about a third of the butter in SMBC, and replaced the rest with shortening. Vegan margarine tastes better than shortening, and a bit of salt is actually good in frostings but you have to add it slowly and have a lot of spoons around to sample that you have just enough margarine in the batch without being noticeable.
Then I would just make some of Alton Browns royal icing for piping. Its mega fluffy when I make it. Like really, really fluffy, if you follow his directions. But if someone was allergic to eggs I`d just blend shortening and icing sugar with DF milk.