Thanks GeneBenson, that did the trick.
As soon as I started the dpkg-reconfigure keyboard-configuration command, I saw that something was wrong in the /etc/default/keyboard file, and I was able to fix it. After a restart, everything was back to normal.
Well, the new MDM behavior partly took care of it; the username is no longer to be typed. Trouble was, my password had one character that is at a different place between FR and US layouts -- though I won't tell you which one
Marking as solved.