I see that most of your issues have already been adressed in the update of the game.
It hasn't been uploaded yet because It's a game jam and I want to wait until the voting period has ended.
Also, about the game over thing, I don't think if I can fix that.
You can jump at the part where you have to climb, but there's a bug I fixed where you hit your head against the side of the platform without touching the sides.