Just to let other people know how it ended: I gave up.
I tried to reinstall all of the packages (node, npm, cordova and ionic) a few different times, following the procedures suggested by a few different bloggers. No way.
Cordova still does not work on my Linux Box and Ionic cannot build any app for Android.
Ionic and Cordova still complain about missing files and insufficient permission but… no matter how you set the permissions and how you install the required files this problem does not get fixed.
The best resource/help I was able to find was this one: https://blog.nraboy.com/2014/09/install-android-cordova-ionic-framework-ubuntu/ (and a few questions/answers at stackoverflow.com).
I think other people should try the installer shell script created by Nic Raboy and should follow his suggestions. They worked very well in most cases.
I will try again as soon I will have another PC available (or as soon as I will upgrade this one with a fresh install of a new Linux distro).