Do you have to pay the Corrupt Official's resource cost to use his damage mitigating effect from your hand?

Text: Interrupt: When a [scum and Villainy] objective would be damaged, discard this unit from your hand to cancel any amount of that damage.

no, you just discard him from your hand

You only pay a cards cost when you Play it. Discarding is not playing a card.

Another question about this guy: does his ability stop any amount of damage total from any number of units/unopposed? Or is it just one of those?

For example, if LS attacks with two Y wings and has the edge, do I only get to stop 2 blast damage or can I stop 4? What about unopposed bonus?

Not sure if there has been a rule clarification on this, I couldn't find one.

You can only stop the damage of 1 of those units. You also can't stop the damage of a unit AND unopposed, it's either or. as damage is dealt, you can cancel one instance of damage.

so in your example, you could stop either one of the y-wings damage, or the unopposed damage.

Thanks, Boba. Is this in the rules or been clarified somewhere or is that just your answer?

yes, interrupts are explained on page 24 and the FAQ on pg 4. the corrupt official's ability is an interrupt with a triggering condition. the triggering condition being damage dealt to a S&V objective. multiple units and unopposed damage being done is multiple instances of damage, and an interrupt is only usable on a single triggering condition.

for example if 2 Y-wings, one with an astromech droid, attacked a S&V objective with no defenders an edge battle would commence, with the LS automatically winning. the LS would have the first strike, so they strike with a y-wing for 2 damage. there is your triggering condition. if you decide not to use it at that time and wait for the one with the droid 2 damage is then placed onto the objective, and play continues. then you check to see if any DS units are eligible to strike. If not you move back to the LS units, and the LS then gets to strike with the second Y-wing with the droid and does 3 more damage, so now you decide to interrupt that damage with the corrupt official. then you check again to see if the DS has any eligible units to strike, if not you make a check for any defenders still left in engagement. If not, the objective gets an unopposed damage which is then placed on the objective.

You are not able to wait until all the damage is done before interrupting it. the damage happens at different times and is not simultaneous, so once you allow the triggering effect to take place, it can no longer be interrupted.

It would be like casting 3 sith events back to back to back and then trying to react to all 3 with vader. you have to react to each event as it happens. lightsaber deflection is another example. you can't wait till 3 seperate instances of damage are done, and then decide you wanted to deflect the second instance of damage, it must be done when it happens, same goes for the corrupt official.

Boba's answer is definitely correct.