Pretty nice proof-of-concept Metroidvania. The Metroidvania aspect was executed well with all the exploration and decent level of non-linearity/backtracking involved.
The fall damage is weird, you die if you fall TOO far but you can also kind of negate the fall damage if you jump immediately after hitting the ground? Also agree with what Aria said about climber gloves, I would've liked it if you had to "grab" the wall by holding an arrow key to climb it or something.
By the way, you can extend the double jump by double-jumping onto a wall to grab it, then wall-jumping, at which point your air-jump is refreshed. Not sure if that's intended.
But honestly I think what you've got right here is perfectly workable. Good work.