Thank you all for your comments and help!!
Let’s start this adventure 
I played a bit with the keyboard Layout Editor and I think I have been able to generate a simple keyboard similar to what I need.
Then I pasted my keyboard generated raw data into Swillkb and got a nice plate.
Here it is what I have been able to do:
Now I imagine I should find a service where I can print print this plate. Is that right?
What’s are the next step? I would be happy to read any tutorial you suggest.
Regarding the keyboard case, how does it work for a not standard/common layout like mine?
The only issue I see would be finding a microcontroller/dev board that has enough IO pins to handle 5 rows and 20 columns without having to multiplex or otherwise complicate the matrix further.
I think, at least for a first version, 82 buttons would be enough. I don’t need LEDS (as usual, not in this phase). To be able to generate midi output would be nice of course, but I’m afraid that as a first project it would get too complicated. What do you think?
On the other hand, it is necessary that the keyboard is able to correctly handle multiple buttons keypress (ideally up to 5 buttons whatever they are).
@huhsh you’re correct. The project Chromatic Button Synth is similar to what I would like to do. His project is a “standalone accordion musical keyboard” (it doesn’t need a pc to work). I would like to make something simpler and cheaper (using my pc to handle sound etc).
P.S. I know how to solder even if I’m not an expert and I have some programming experience.