If you access it again by hitting enter from browser address bar, not using refresh, will you get 304 response?
I wonder if the request was served by rails dynamically.
check the MIME type of ".png", it should be in "Expires By Type" configuration
http://www.litespeedtech.com/docs/webserver/config/general/#expiresByType
looks like that file has been served as static html file.
Please check the .htaccess see if something override the MIME type for ".php" to "text/html".
/usr/local/lsws/admin/conf/php.ini is only for litespeed web console, for other web sites, you need to check phpinfo() page to find out the correct php.ini
For temp directory, you can move "Static GZIP Cache Directory" to some where else...
When you run the install.sh, when answer "using Litespeed web server with a hosting control panel?" you need to select "NONE". then use "www-data" as the user.
Please remove /usr/local/lsws directory, then try again.