1. In my personal opinion, I actually preferred jumping around like a madman in your previous build, so I do not like the cool down as much. Especially because you can't tell when you can use it again, it was really frustrating when the lava catches up and you can't do anything because the cool down is too long. If you wanna keep the cooldown I would make it significantly shorter so it doesn't kill the pace of the game, and also add some visual or UI element so I actually know how long the cool down is.
2. I think it makes sense to have some kind of bar that visually shows how much distance is between the lava and the player.
3. As I said above, I think you should just start off with a much faster cooldown, and if the player gets too far from the lava you can have it do some sort of rubber band effect to catch up and keep the difficultly consistent. That way difficulty doesn't rely on the player waiting around to be able use the grapple again.