In any connected Zome structure, all of the balls will have the same orientation in space.
A proof by induction on the number of balls is pretty easy to see. It hinges on the following key observations: 1) The two balls on either end of a single stick have the same orientation. 2) In a connected structure, each ball is connected to some other ball, and then to another ball, and so on.
There are never enough Zome pieces.
The proof of this theorem is by excessive experimentation. The more you experiment with Zome, the more you'll find yourself running out of pieces before you can finish a particular shape.
There is no such thing as "the last dandelion in your yard".
By observation, of course. This has nothing to do with Zome, but I'm a homeowner and it's dandelion season and I just had to vent somewhere.