I have associated these options to the correct command, so yes, when I click stop or start ; my server does that (I can check if the default page is loading or not).
Also, when I stop my server from SSH or restart it from SSH and reload the module page, the webmin module detects correctly the current status of the server.
At this point, I think it would probably waaay quicker to solve this if you would install and test on a box you control (there are free trial licenses). The back and forth here takes a lot of time. I gave my configurations of the wembin module above.
I have also changed a few settings from the litespeed control panel :
In the subsection “Using Apache Configuration File”
Load Apache Configuration Yes[...]
Apache Binary Path /usr/sbin/httpd
Apache Configuration File /etc/httpd/conf/httpd.conf[...]
PHP suEXEC Yes