Am not certain why one would be leary of running mysqltuner ... it's not a daemon (service) ... just a perl script and takes just a minute or two to run. Output, however, worth the minute or two taken to run.
Executing cron will kick in different task at different times... some not so heavy ... some really heavy ... like automated backups. So just executing cron during your work day - after you've had to reboot server may not give you a clue/hint.
Logs .... depends on what flavor of server, but .... apache error logs, php error logs (which might be in moodle code directories where there were issues ... mysql DB error logs. Then there is also the need to interpret those logs ... warnings are not errors ... and errors might/will beget more questions.
Since the server cloned is on/in a virtualized environment, and this happens on a regular basis, think I'd inquire with the virtual server admin - you might have to be insistent as I've never talked to a virtual server admin type that could imagine something wasn't quite right for guest OS's. :|
Also, one might want to network folks about monitoring the system.
Don't think anyone could tell you anything specific at this point.
'spirit of sharing', Ken