Automatic mounting for plugin devices and auto-play options for audio and video disks are in Settings under 'Removable Drives and Media'
Here you can set the system to auto mount and auto play and set specific applications to run depending on if a finished or blank disk is inserted.
To play a DVD the system has to mount the media. This would suggest the hardware is functional at some level.
A good start would be to prove the DVD hardware and RW function works (with your media) by connecting it to an alternative system if available.
Back on Linux Lite, to debug the mounting and hardware detection issues systematically, we need to review your settings and check the logs for events related to the failures you are seeing.
