Before anybody builds anything, I always like to start with the question "Why?", and then encourage people to set out their criteria for the building before attaching any technologies or materials to the project. Then, when a person knows what they want to achieve (and why), it's much easier to decide how to go about achieving it.
That said, I'll assume you've thought this through and your proposed building is what will best suit your needs/wants.
There is no good and accurate way to size an AGS system. There are only a handful that have ever been built, and of those, even fewer that have accurate test results that would help to create some guidelines. Either yourself or with a suitable engineer, you'd have to look at the heat capacity of the particular soil you're working with, and the Btus you can expect in a typical year from whatever sort of collector you'd be using. You'd have to guess at the system's inefficiencies and losses, and you'd arrive somewhere in some ballpark that might hopefully be reasonable.
Chief among the losses to consider is migration of your stored heat away from underneath your building. The ideal AGS system would be fully excavated, lined with a really good insulation, and then have the dirt returned to its insulated home. Without full insulation, losses downward and to the sides are substantial, as you are essentially trying to warm the planet in every direction under your home, and the planet has an almost infinite appetite for your heat. That's a lot to try and figure out accurately...
Arched, thin shell concrete has some great structural properties and can look great from the inside. Again, I always try to return to goals and criteria. Why this kind of ceiling? If there are compelling reasons, great. The thin shell concrete will not offer any amount of insulation value, so you'd have to not only figure out the roof design and green roof layers, but be sure to be including adequate insulation for your climate. You may have read that green roofs provide a reasonable R-value, but this isn't really the case. They definitely help keep a roof from overheating in the summer, but in the winter they offer little to no insulation value. An undulating roof like the one you are proposing would be difficult to insulate in any way other than spray foam. Nothing wrong with choosing spray foam unless concern for the environment is anywhere in your criteria...
The edges of such a roof assembly will be difficult to finish cleanly, and it will be hard to keep runoff from coming down the walls and staining them. Straight edges can be troughed, but round roof edges make it hard to collect/direct water.
There are many companies with mortar sprayers that you can hire. You can even buy a decent mortar sprayer (we own a Tirolessa http://www.tirolessausa.com/
), but you'll still have to hand trowel the sprayed plaster unless you like the random, fuzzy, bumpy look of sprayed plaster.
Your project sounds interesting, could be feasible, and also seems like a lot of work! I would suggest making sure that you are getting what you want out of this design before committing to it.