I would like to have the left and right on the d-pad instead of the left joy-stick, can I do that, and if I can, how?
If one can't setup controls for this game I guess I'll have to use the keyboard...

I use JoyToKey since my game doesn't notice my controller (or even remembers settings...).
You'll just make a profile on JoyToKey to sync your controller buttons to the game. After that you'll just need to start JoyToKey with the game to play normally.

Looks a little bit confusing, but I'm gonna experiment with it for a while and see if I can get it work out...

It works, but I'm not sure if I've all the buttons... Is there any more controls than left, right, up, down, jump and attack?

Enter is use to pause the game and BackSpace is to leave a level when you've complete it. Besides that, you're good.

Thanks, I had a feeling that there was something missing.

Also just adding to the pile you can use xpadder as well.

It seems like I'm getting problems with doing the wallclip with controller. Does that only happen to me?

It's not an easy trick.

The usual method is to either use the keyboard keys or bind Left and Right to two buttons, like the shoulder buttons. As you're moving towards the wall (let's say to it's the right), let go of right and press left, then hold both buttons as you're about to hit the wall. The faster you press the button order, the more momentum you'll keep.
You can do it with a d-pad by flicking the opposite direction while moving, but the time frame is very small.

Is there any way to have the nintendo joy-cons compatible with the controls?