Can I command/activate a unit that has already been exhausted?

For instance I had already exhausted Vader, the Rebel chose to Heal which triggered an upgrade card. Something along the lines of "after a rebel hero heals, you may interrupt to move or perform an attack against that hero." Can I have Vader attack or must I choose someone who hasn't been activated yet?

Yes, this is legal.

Exhausted units have merely spent their own activation, and can be "ordered" for example, by Imperial Officers or Gideon Argus, to perform additional actions.