Do pilots ever use the aircraft's ICAO 24-bit address when communicating with ATC?
Typically pilots do not know the 24-bit aircraft address of their aircraft.  

The 24 bit address is intended for data protocol level of communications and is not used in voice communication. Therefor there is no need for pilots to be aware of their ICAO 24-bit aircraft address. They use the aircraft's registration if they need to identify the aircraft they are using for the flight. 

The 24-bit address is used by transponders for communication with Mode-S radars, ADS-B transmissions, by TCAS for tracking traffic and coordinating Resolution Advisories and by Data-Link communications.