Taxing my memory here about 3.old ... :|
A guess: your custom plugin doesn't inform the regular cron job/task of jobs it shouldn't run ... like 'clean up' of moodledata/temp/backup/ ?
On restores that fail, what is left in /moodledata/temp/backup/?
There should be long directory names comprised of a mix of alphanumeric characters. Contained in them are the parts of a course moodle is to restore. Uses xml files and related directories for things like files ... think there should be a questions.xml file.
Around 3.6 all assignment modules should have been converted to the new 'assign'. There was an additional Site Admin Menu item to accomplish that ... by 3.11.x think that went away.
In your future marched versions ... an attempt will be made to auto remove assignment directory in mod (you see it in admin as assignment (2.2)) ... if I recall correctly the advice was to unintall assignment (2.2) prior to any attempt to upgrade to series 4.x. Ditto for question types examview.
'SoS', Ken