Doesn't that version of PHP come with Zend OpCache?
php -m |grep Zend
Would show if the module is loaded.
Do you have any tool for checking OpCache configuration? ... like OpCacheGUI, or other. This might help if you do:
http://stackoverflow.com/questions/17224798/how-to-use-php-opcache
Can you get to the Site Admin Menu?
Check the assigned roles of one account that is NOT affected. Also check the role of an account that IS affected. Do you have any custom roles for teachers? Then check:
Users -> Permissions -> Then one of two tools: Check System Permissions or Capability overview.
Can login as one of those affected users ... not your admin account.
Noticed in the screen shot the menus are docked. Can a user who cannot add an activity un-dock those menus and then try adding a resource? Themes have to support docking.
From the setting is says:
"If enabled and supported by the selected theme users can choose to move blocks to a special dock."
Any customization to theme being used? What happens if one changes theme?
Those settings per user would be in sessions ... file or DB (although criptic ... ie, I could not tell you what value in the sessions info was for permissions to what ability).
Since the site has been in use for a long time, what of checking tables to see if any are in need of repair?
mdl_role_capabilities???
'spirit of sharing', Ken