As for the egg I found this on a vegan site
1 Tablespoon flax seeds plus 3 Tablespoons water replaces one egg. Finely grind 1 tablespoon whole flaxseeds in a blender or coffee grinder, or use 2 1/2 tablespoons pre-ground flaxseeds. Transfer to a bowl and beat in 3 tablespoons of water using a whisk or fork. It will become very gooey and gelatinous, much like an egg white. In some recipes, you can leave the ground flax in the blender and add the other wet ingredients to it, thus saving you the extra step of the bowl.

For your frosting I would just use powdered sugar, a dash of vanilla and enough soy milk or soy silk creamer to make is spreadable or drizzable whichever you prefer.

I would strongly suggest using a tablespoon or two of earth balance in the frosting... it gives it a very rich flavor.... and the salt... but just a dash.

the best tip i've learned recently is that salt goes in EVERYTHING.

And if you use enough powdered sugar, the cookies shouldn't get soggy at all.