Does that attack cause one damage to the objective for being unopposed???

Yes. Unopposed damage is independent of icons or even having a unit able to strike. All that matters is that the attacker have at least one surviving unit and the defender have none.

Edit: Never mind. I said something silly. Xenu's Paradox is correct: if there is a surviving attacker and no surviving defender, no matter what other circumstances, you'll do 1 unopposed damage.

Yes, you may want to put this sort of thing on the Rules forum though.

Another question.

Lets say you have 5 units attack an objective but no defender. Do they each cause one damage or does the whole flurry cause one damage?

Only one. You check for survivors, and if there are any on the opposing side (and none on the defending side), you do 1 damage, regardless of how many units survived the encounter.