Get a lamp from the house. You can check all of the 120v AC receptacles with that. This part of you power is controlled with circuit breaker just like in your house. You need to be plugged in for this side to work.
You will have a converter that gets power from the 120v AC side and converts it to 12v DC. This side uses fuse's like in your car for control. It also charges your battery. The battery may have a cut off switch to prevent draining power when in storage. This is what I would look for before buying a battery.
If your fridge is one of the automatic types it will run on 120v AC (house type) when plugged in. With out being plugged in it will need the propane turned on and the 12v DC battery charged for the control circuits.
The water heater will run on propane and maybe 120v AC if so equipped. The red switch probably has a indicator light but the propane has to be turned on. You will hear it light, it roars like a jet engine. I won't hurt to turn it on for a few moments to verify that it works. It also needs 12v DC from the battery or the converter to ignite.
The furnace uses propane for the heat and 12v DC for the fan. One battery on a cold night may not make it through a night with out being plugged in.
You need to get a cheap volt/ohm meter to be able to check battery voltages. The indicator lights in your bathroom are notorious for not being accurate.
