I had a good fun with this one! The character design is cute and their non-wall-jump abilities are a neat take on typical staple mechanics. I think the total resources put in were fairly well distributed, with nothing really standing out as having been left behind entirely. It sets up promise that some more polish could make this game shine in a well-rounded way.
The overall character control is probably the first thing that can benefit from some tweaks, as it is the most prevalent factor with this platforming focus. On controller, it'd be nice to have a shoulder button input as alternative for the basic jump, as a lot of the navigation benefits from having constant camera control.
Also on controller, the dash's sharp turn radius makes it a bit hard to control, as being slightly off from straight ahead makes you zoom off of the path quite easily. Having the camera snap behind the player while dashing seems like an overall good call!
I think the jump and fall arc could use a bit sharper of a curve at its apex. It can be hard to tell with the floaty jump, especially when platforming up, when you will hit the ground. This is particularly noticeable when going up via falling platforms, as you want to jump pretty immediately. When you get to do that jump is a bit muddy, and your floatier air controls turning into the sharp ground controls at an unexpected moment can send you jumping off-target rather easily. My initial instinct would be to have a set time at the apex where you just float and do not land, after which you get a sharp initial burst in downward speed, to serve as a smaller time window in which the player would land, while preserving the overall arc of the jump.
Dashing into high-jump makes for a very fun joust-and-pole-vault combo, cohesive and uses the character design well! I think it could benefit from having dedicated transition animations, sound effects and all that, and possibly some fancy momentum-redirecting based on how the vault is executed. Working the staff into the wall jump animation would be neat too. Would love to see these abilities polished up, expanded upon and tested in conjunction in some flowy, momentum-preserving platforming sections.
In level design, I noticed there are some slight verticality differences in certain platforms, such as the wooden ones. These really don't need to have collision on these differences, they can be just one rectangle for logical purposes, for smooth play and and performance. The wall jump ability, unlike the other two, seemed to not come with an immediate test challenge to verify the player understood the admittedly informative (and endearing) wall painting.
A quick win on the audio side would be to add (some variant of) the stepping sounds to when the character lands. That should make it a bit easier to grasp when exactly you regain your ground controls, as well as just feel more correct. Some particle effects alongside that would be a nice reinforcement.
Little performance note: The game would stutter a bit on initially playing in the starting area, both in the web and windows versions.
Good stuff, keep it up!