1) I feel like the cool down made the game more enjoyable. I agree with previous comment that it was a little frustrating at first, but after a few rounds I got the hang of it. It seems better than your previous version where you could just click like a mad man to shoot up the level. Definitely more challenging.
2) I think showing maybe a small mock up of the volcano off to the side with indicators of player and lava position could be a good way to visually show how far away you are from the lava. A number of total distance climbed would also be nice and also give a good metric to display at the game over screen so players can see how well they did.
3) I think a small scaling down of the grapple hook cooling with lava speed would work but may run into issues if the player can just get really far ahead. Also I didn't notice any indicator for the cooldown or when the hook was available again, so maybe adding something to visually represent that would be nice.