I’ve done a fair amount of googling, and I can’t for the life of me figure out how to change the “off” and “on” switch actions to a different mdi icon. Basically wanting to change them to mdi:arrow-up and mdi:arrow-down.
You cant. In the same way that you cant change the icon of a toggle switch. These icons are built into HA for switches that do not have state feedback.
If you provide a valid state_topic: they will change to a toggle switch.
This will also occur if you set optimistic: true for the switch - but be aware if using this method the switch in HA could get out of sync with the physical state of the device if something other than HA is used to control it (an IR remote for example). Also a toggle switch isn’t much use for a volume control.
Your other alternative is to use any of the built in or custom button cards to control the switch. e.g.: