3-Phase Theory vs Practical example confusion
"Wye obviously containing a neutral line"


Of course not. You connect only three phases without neutral, the ground goes to case for safety. $$i_a+i_b+i_c=0$$ 
The summ of all currents is zero.

The ground is mandatory, there is no such possibility without having the earth/ground.

The casing is isolated from windings, no current flows from earth/ground to the phases. The ground is just for safety purposes. 

A three phase motor can't run on single phase.