Once again, I think it is a matter of chain of effect from invisible creature to damaged creature.
Invisible Imp can cut the rope on the bridge and kill the PCs. He remains invisible because he interacts with [rope], which does not in any way damage the PCs.
Invisible Imp can open portcullis to release attack dogs or summon creatures to do his bidding. He remains invisible because the decision to attack the PCs belongs to a third party [dogs] or [summon].
I think the key word in the entire description is REMOTELY.
The reason i think the Imp would lose invisibility in the case of the falling statue is because Imp interacts with [statue] which immediately falls and damages [PC/dwarf]. You can draw the cause of the damage as a direct line from Imp to PC, with the statue being in the middle.
Firing the trebuchet would require the Imp to directly interact with [trebuchet] which immediately fires at [PC]. There is a projectile that flies directly from [trebuchet] to [PC], which was fired by the Imp.
So Imp -> [trebuchet] -> [projectile] -> [PC]