I'm not sure what the use of Pierce 1 is on the Probe Droid card. Aside from the possibility of getting two surge on a die, I don't see why I would use this ability over the option to do an additional damage. Pierce 1 seems like an inferior version of +1 damage. Tactically, is there a circumstance when an Imperial player would choose Pierce 1 over +1 damage?

No reason to choose it over the +1 damage. It will just get used when you have extra surges.

I haven't looked at all the cards though - it's possible there is something in there that gets triggered if you use Pierce, but I haven't seen it yet if it exists.

If you do get a second surge it will be useful when they cancel one of the surges and since you roll two yellows getting two surges in an attack should be pretty common.

So are you allowed to spend multiple surges on the same thing? I couldn't find in the manual where it says so one way or another.

Nope, one surge, one effect.

Once an effect is used, it can't be used again on the same attack.

When do players declare when a surge gets canceled? Before I choose my surge effect or after?

Before. If you roll 3 surges, and they roll 2 surge cancelers, then you have one surge remaining to spend on a surge skill or restore fatigue.

Pierce is - as of now - a lesser version of +1 dmg, although it often does the exact same thing and on models that have both (like the probe droid) it allows you to double-up +1 dmg when you otherwise couldn't.

My guess is... There is some mechanical 'wiggle room' in there for some rule interactions later on in the game, in the form of 'if this attack does 1 or more damage and your opponent did not block, then _______ ' This would provide the edge to Pierce over +damage.

Well, if you role two uncalled surges and they didn't roll a damage canceled you could only get plus one damage. Being able to get two extra damage might be excessive. Seems like a balanced card.