Appears we are involved in prep for upcoming new semester which is usually a busy time for moodle servers. Teachers/admins doing things not normally done throughout the year.
Investigate 'context ID' ... many quizzes look to be from a couple of courses. Find the course and investigate the quiz banks for those courses.
reference in php error logs to a moodle.domaine.fr instance of a Moodle. Is that obscured for security purposes?
Do you have multiple instances of moodle on that one server?
deletions of modules in courses
Set recyclebin time to retain for 1 second to hurry along deletions to trash.
Set recyclebin link to show all the time in course menu so with troubled courses that have deleted modules you can investigate.
PHP logs
MySQL server has gone away
Indicates the need to tweak some DB settings upward.
Is DB server localhost?
Suggest installing mysltuner.pl and run with superuser creds for the DB server. Pay attention to:
Performance metrics (information on memory) as well as InnoDB metrics (InnoDB buffer pool / data size)
'SoS', Ken