Our Pod's water heater only operates on propane, though you do need either shore power or the batteries to be connected in order for it to work. When you turn on the switch, it turns on the water heater which will then run until the water reaches its set temperature. It kicks off then until the water temperature falls again.
Your lights, water pump, and furnace will all run on battery power, as will your slide and your refrigerator (not recommended to run it on 12 volts; run it on either shore power or propane, but remember it consumes a little battery power even on propane mode). All of these devices work on 12 volt power regardless as to whether you're connected to shore power or not. When you are plugged in, your converter takes the 120 volt AC current and converts it to the 12 volt DC current these devices need. And yes, it charges your Pod's batteries at the same time, but be aware that it will generally take hours to fully recharge your batteries.
Your AC outlets, air conditioner, and microwave will not work at all without shore power (or generator power).
There is a lot that can be said about batteries, but suffice it to say that you shouldn't let them drop below 50% charge (or capacity, same thing) before recharging them. A 50% charge is equal to about 12.2 volts. You can monitor the voltage will a voltmeter attached to the batteries or with one you hard wire into the Pod somewhere, a very common modification.