First, thank you. Our son is GF, DF, NF, and is always the one left out with school snacks/desserts. You're doing a great thing being inclusive.
Sticky Fingers bakery, in Columbia Heights, has cakes or cupcakes that are GF, DF, NF, and they're fabulous. Close your eyes, taste them, you'd never know the difference.
It's worth buying instead of trying to bake this yourself. Some Whole Foods also have Sticky Fingers stuff in their bakery sections.
If you're in Bethesda, Lilit Cafe also has GF, DF, NF desserts from a great bakery in Baltimore.
If you do decide to do it yourself, I can't help on the baking front, but for frosting, Duncan Hines in a can is GF, DF, NF. Better confection through chemistry.