Can I use Zillo Technique after my opponent spends surges, of do I have to use it before?

The top part can be used just before step 7. Pierce gets used during step 7. Pierce does not remove block symbols, it lets you ignore blocks during step 7. (We got a ruling for Heavy Armor which suggests so - see https://www.boardgamegeek.com/thread/1545259/command-card-heavy-armor/page/1 .)

The bottom part can be used during step 4.Apply Modifiers (before 5. Spend surges). (If you use it later, the block is not added to the defense results.)

Great, thank you.

Side note:

Can you use the bottom part if the card is exhausted? I'm pretty sure you can but just checking.

Yes. The bottom part does not have a cost, so it is always available as long as the card is not depleted. (Not that Zillo Technique could be depleted.)