Can android devices be controlled/tracked (including on/off control) by IMEI?
YES Any phone could be tracked and localized using its  IMEI  but to successfully do it the stolen phone should be online by:

Internet connection
Sim card inserted

But if the guy who got your cousin's phone could change phone's IMEI(and for sure phone identity) : Sorry to tell you that the phone is gone forever.
The best solution of all times: daily cloud sync.