Well, by popular demand (and my own curiosity) I weighed this beast.  I was a bit surprised by the total - without furniture - came up to be 84 lbs (38 kg).

For transporting the structure, I plan to get some plastic storage containers that can hold a single level in them.  I will then lay the containers out covering the floor of the vehicle and then other terrain can be stacked on top of them.  When storing the structure, I'll likely keep the levels stacked together and covered with a cloth to ensure they don't warp and will continue to fit together properly.