Categories and Menu-structure gone from dashboard but still work in front-end

I’m building a website for a client (for the first time) and like the title says, all categories are gone along with the menu-structure in the control-panel/dashboard/whateveryoucallit. Everything still works in the front-end, however, since these taxonomy-items aren’t available when adding or categorising content, I can’t post anything. The menu still works but appears to be completely flat in the menu-settings – all items are there but without the hierarchy.

The site is here https://helsinkicommittee.dk/ however looking at the front-end wont provide any answers, I think.

I’m using a twenty-twelve-child theme and I’ve tried switching to twenty-ten, disabling all plugins and re-uploading the wp-admin and wp-includes folders as well as the wp-content folder without removing the new files of course, but nothing has changed.

Others have told me it is because of the latest wordpress-update, so a support-guy from my hosting-service (UnoEuro.com) told me to downgrade to a previous version of php by adding “AddType application/x-httpd-php-old .php” to the bottom of the .htaccess file but this made no difference either, and now he’s asked me to take it up with a wordpress forum instead.

Does anyone have a solution to this problem? Others seem to have had the same issue through the years on this forum, but I haven’t found any solutions that worked for me among them.