johnny mkdir(): Permission denied
Your PHP simply can't write to disk, which is necessary if Files app is to store config, resized images and folder cache. I can't diagnose this from here without access, assuming this is a private server. Normally, it would be permissions, owner mismatch or your website document root pointing to areas on disk that PHP may be able to read, but not write. You will need to ask in forums for the specific server combo you have setup.
Using Files app without cache
It is possible to edit the config in index.php
and disable all cache features, but this is slow and inefficient. With cache features disabled, Files app will no longer require capabilities to write to disk.