Cool entry!
Gameplay wise, I think you have a foundation for a solid platformer, with lots of things you could do to expand on it! However, I think the execution could use some more work, as the controls feel a bit slippery and I often struggled to do things. I feel like the movement in general is too fast and the gravity is a bit too high, so things feel a bit too "twitch reaction" but the game doesn't fully give you the control to handle it. There is currently no indication for when you're in range for the pickaxe to work, so sometimes you just jump and press LShift and hope it'll work. Because of the level design, where sometimes rocks are placed in areas where you'll die if you fail the hook, this can feel a bit frustrating. Maybe you could add something like an animation for the character or a glowing outline for the rocks if you're in range, so that the player has better feedback. This, coupled with a slight rebalance of the speed, could make it so that the player in general feels more in control. I'm not sure if you're using Unity's Rigidbodies and AddForces for this, but I find those may feel a bit inconsistent for a platformer unless you tweak them just right with gravity values and stuff. Some people even implement their own gravity functions so that you hold a bit longer at the apex of your jump, and fall faster after that. This game by GameMaker's Toolkit is an amazing resource to learn more about platformer controls, if you'd like to know more about it. You could also maybe play around with the idea of having the pickaxe hold you in place until you jump, so that the player then has a way of controlling the second jump a bit more and you can get tighter controls and platforming challenges.
Thematically, I feel like the connection is made in the mechanics and gameplay, but not too much in the presentation. You could add some things to make this feel a bit more connected, like, for example, having a fire icon next to the heat bar, and a red border appearing on screen if you're close to overheating. You could also maybe play around with some lighting effects so that the cave appears darker and the lava is what emits light!
In terms of the aesthetics, I really liked the art you had for the story bit, and how the character pops out of the wall at the start. I think there's definitely a lot of care put into these parts! In general, I'd suggest working a bit more on the visual feedback side of things, so that the game feels more responsive. An example would be the aforementioned outline around rocks when you're in range, or the visual effects when you're overheating.
In terms of the sound, some parts could use a bit more sound effects. The start of the game where the player pops out of the wall is cute, but it feels a bit empty because there's no sound effect for when they're shaking and struggling to get out. The music and other sound effects feel good though!
I hope some of this feedback can be useful to you! It's all just my subjective opinion, but still. Keep up the good work!