Jump
Pressing JUMP sets your vertical speed to 10.5, and it will decay by -0.5 every frame due to Gravity. If you let go of JUMP while still going up, Peppino's vertical speed will instantly be set to (almost) 0, resulting in a variable-height short hop. Peppino's horizontal speed will neither increase nor decrease while Peppino is in the air, meaning Jump is generally safe to do while Mach Dashing, but will cost time if Peppino is not already at max speed, as he won't be accelerating.
Jumping actually has 2 states: Walking Jump and Running Jump. Peppino performs a Running Jump if you are holding DASH while Jumping at any speed, and a Walking Jump if you are not. All other actions that put Peppino in the air (walking off a ledge, bonking on a wall, letting go of air run, etc) will put Peppino in the Walking Jump state.
The Walking Jump state is actionable into Grab, Taunt, Bodyslam, and Uppercut. The Running Jump state is cancellable into Grab, Taunt, Uppercut, and Dive, and Peppino will start Wall Runs when he touches the wall, transferring his full horizontal speed into vertical speed.
Long Jump
Pressing JUMP during a Grab will prompt Peppino to lunge forward. Peppino is considered to be dashing during this state, which enables him to Wall Run if he touches a wall or link to a Mach Dash if DASH is held as he touches the ground.