Televisions run on 115 VAC, so if you disconnect from shore power they will turn off.
My answer to the other question is from my experience with the AS trailer.
When plugged into shore power, your 12 volt
lights and other 12 volt
accessories (stove light and fan, Fantastic ceiling fan) should work, regardless of the use/store switch. If the switch is store, the converter will not charge the batteries.
When NOT plugged into shore power, if the switch is set to store, nothing will work except the propane detector. If the switch is set to use the lighting and other 12 VDC devices will work if the battery is charged.
All of your 115 VAC devices (television, A/C, heat pump and the 115 VAC wall plugs) will work as long as you're connected to shore, regardless of the use/store switch.
For your motorhome, I don't know how the house battery works with the vehicle battery.