You need to setup the add-on AND the integration. If you go to Settings > Devices and Services you can view a list of all the devices home assistant can interact with and control. If you don’t see something like the following then you didn’t setup the integration.
This is incorrect. Are you mixing up add-ons and integrations?
If you are using HAOS, then you will install one of either the Z-Wave JS or Z-Wave JS UI add-on. If you are not using HAOS, then you install something like Z-Wave JS UI yourself (container, binary, etc.). Those provide the Z-Wave JS Server (1), which HA connects to via the Z-Wave integration (2).