I'm honestly not entirely sure if diagonal movement would be such a good thing to add. It didn't exactly make the controls in Brutal's Challenge DX easier or more intuitive. And due to the way button input detection in RPG Maker works, it could throw players off severely when they're trying to quickly switch between horizontal and vertical movement. Besides, so far it didn't seem like diagonal movement would really be necessary in this new game to begin with.
Unless you decide to implement an option for the player to turn diagonal movement off or on entirely, I don't see how adding this feature would actually improve the experience as a whole.
The other ideas sound good, though. Maybe it would be useful to add a small reminder image to the UI when a player has collected enough cool points to jump, showing the amount of available jumps and the button that needs to be pressed to activate those jumps. If you keep it small and place it somewhere on the bottom of the screen, it shouldn't overlap with anything too much.