Where and when does firmware of a device run?

Does firmware of a device run in the device or the device's controller?

When does firmware of a device run? Does it run if and only if the cpu sends commands and/or data to the device's controller?

For example,

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  for a hard drive and its controller and firmware, does its firmware run in its controller and if and only if the cpu reads/writes to the hard drive?

  
  


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  for a wireless adapter and its firmware, does its firmware run if and only if the computer is connecting or connected to the wireless internet?

  
  


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  for BIOS, is it firmware for the bare computer system. Does it run in the cpu? Does it still run after the OS fully started?

  
  

Thanks