1.) I think the cooldown on the grapple can work, but there are two things going against it in my experience. First, the random level generation can be frustrating to navigate with in some configurations. In some places I felt like I couldn't grapple to where I needed to go very easily, and being under the time pressure of the lava made that feel a bit frustrating. Second, I couldn't tell when I would next have the grappling hook, which made me jump to my doom a few times. On the plus side, I felt the cooldown made me think about how I was going to grapple a lot more than in your previous prototype.
2.) I think either way you mentioned would work fine. Personally I think I would prefer a number, perhaps near the bottom of the screen. I think on top of that a number showing the distance you've climbed so far would also be good.
3.) I never got to that point (at least I don't think I did), so I didn't experience the quick lava. That being said, I think changing the grappling hook's cooldown would make it very frustrating to know when it's available. Instead, I'd recommend thinking of some way of slowing down the lava instead. Perhaps some sort of collectable?