Each of the six 135kW chargers draws power at 480V phase-to-phase, which is supplied by the low voltage side of the transformer. Even at 480V, the total current drawn by all 6 chargers is so high you want to install the transformer close to the SC cabinets. The cables being pulled off the large spools will be carrying the high voltage from the original HV vault to the transformer pad. A common distribution voltage is 12KV, but I understand newer distribution voltages go as high as 35KV.
Absolutely!
Most of the thickness in those cables coming off the spools is insulation to take care of the 12-35 kV.
The current draw on the Medium Voltage side of the transformer is pretty modest and does not need much wire thickness.
1,200Amps*0.48kV/12kV = 48 Amps and 1,200Amps*0.48kV/35kV = 16 Amps
That was one of Tesla's great ideas!