About the error logs you need to check in your hosting account or contact the hosting support, because it can be different from hosting to hosting.
Maybe some wrong setting in the W3Total Cache is causing it.
Can you try to change temporarily the htaccess(make a backup of the existing one) replacing it by WordPress Default’s.
# BEGIN WordPress
<IfModule mod_rewrite.c>
RewriteEngine On
RewriteBase /
RewriteRule ^index\.php$ - [L]
RewriteCond %{REQUEST_FILENAME} !-f
RewriteCond %{REQUEST_FILENAME} !-d
RewriteRule . /index.php [L]
</IfModule>
# END WordPress
-Rui