The issue/problem is no longer occurring after doing these steps below. I tested your site with Boom Proxy and 403 errors were no longer being logged.
1. Click/Select the Delete plugins htaccess File Radio button.
2. Click the Activate button to deactivate/delete the Plugin Firewall htaccess file.
3. Click the Save Whitelist Data button.
4. Click the Create Firewall Master File button.
5. Select the Plugin Firewall BulletProof Mode Radio button.
6. Click the Activate button to activate the Plugin Firewall.
The php error has to do with Ownership of the /languages/ folder. You are using open_basedir but the /languages/ folder is in the allowed/unrestricted paths that you are allowing for writing. You will need to check that the /languages/ folder has the same Owner as all of your other WordPress folders. It will be different since a writing problem is occurring so you will need to change Owner of the /languages/ folder to the same Owner as all of your other WordPress folders. It is possible, but not likely that this is a permissions problem on the /languages/ folder so you should also check folder permissions.