“Persistent: Each time Paldeyn would gain a Corra, you may instead immediately deploy a Lyon, Ryder, or Ironwill.”
If you would gain multiple Corra at once, does this trigger for each Corra? (no)
If you would gain more than 1 Corra from a single effect, this ability still only triggers once, as it is one “instance” of gaining Corra. As well, you must forego all Corra you would have gained from this instance in order to trigger this ability.
Can you trigger this ability from stealing Corra? (yes)
This ability can be triggered from stealing Corra, as stealing Corra is gaining it. However, if you trigger this, it replaces the steal and the opposing faction gets to keep the Corra you were going to steal.
Do you have to pay the cost of the elite you are deploying from this ability? (no)
The cost of deploying the unit is in foregoing the Corra would gain. You do not need to pay additional Corra, even if the cost of the elite you deploy would be more than the amount of Corra you would have gained.