Yep, the CNC stuff works now, more or less. After a long night at the studio, i managed to get the thing to do what it's expected. So, it not only moves controlled by the computer, carving out stuff, it does it with at least some degree of accuracy - a supposed 10 cm move is a real 10 cm move, also according to the tape measure. Also, i found the right combination of software to make the machine accept GCode, the standard format for this type of machining.
The board i used for this machine is Arduino Mega 2560. It's bit of an overkill for this task, so it's not too surprising that most CNC software is originally written for lesser (and older) Arduino versions. There's also quite a lot of software for 3D printers out there, which use the same principles as this machine (XYZ motion, GCode instructions), but with the addition of controlling the extruder head which is kind of a hi-fi version of a hot glue gun, and other extra stuff. I tried those too, but honestly my skills (and especially patience) were not up to the task of bypassing all that excess code to make the software work with my crude barebones machine.
I don't actually remember the exact combination i finally ended up using, but i'll try to document that when i get back to the studio. Also, some coding was needed to make my machine work: some basic configuration (i.e. how many rotation steps make one mm of movement) as well as bypassing some code (i do not have end stop switches on the machine, for example).
After all this, the machine works, as said. But just barely. I can already see dozens of improvements that need to be made in order for this to be an actually usable tool instead of just a fun curiosity project. And as far as i've read other people's experiences, that is an endless road. Many people end up doing multiple machines, each better, more capable and accurate than the earlier ones. I wouldn't be too surprised if that happened to me too...