I installed joomself free edition, which worked quite okay!

But I'd the bad idea to trash a folder… and all went wrong! Gosh…

Then I tryed to reinstall: purge the db, manually trashed all I saw with joomsef name, and one more installation.
Impossible! I get the message (sorry for the french):
Erreur de construction des menus de l'administration
Installation d'un composant : La fonction DB ne rapporte aucune erreur
Échec de la suppression de joomsef.xml
Échec de la suppression de joomsef.php

What did I make wrong?

Probably you deleted something you was not supposed to delete. Seems like the JoomSEF plugin is missing.
I would recommend deleting all related to JoomSEF and reinstall again from scratch.