Liquid cooled charging cables is pretty standard for high speed DC fast chargers already. When that cooling is broken at an Electrify America charger, for example, charging is limited to something around 50kW.
Liquid cooling is indeed commonplace among DC fast charging systems. The Tesla V3 supercharger cable is also liquid cooled. The novel aspect of the Tesla V4 supercharger cable is that the conductors are directly immersed in the coolant.
Unknown. An interesting data point, however: Tesla's "V4" charger uses liquid cooled (!) conductors to supply 1KA at 1KV; 1MW.
I can't explain where they're getting that supply, but apparently they're doing it.