Turns out the fire was a dud - you can't use the fire to burn the rope. Instead, the fire is there to limit the amount of time you can use an item to jam the plank open. Using the other propeller, which is fireproof, I jammed the plank open, then used the thief's explosive arrow to cut the rope.

Which platform are you playing on? If you're playing on PC, they don't do anything yet, but Frozenbyte has stated that they will in the future. On Wii U, it unlocks the exclusive level. UPDATE: Trine 2: Complete Story now uses them! Just like the Wii U version, they unlock the Dwarven Caverns level.

There is a catapult in the first level of The Goblin Menace, just after you go outside the tavern and do the 3 platforms puzzle. Use the lever to operate it.