It sounds like that water heater is a lp gas water heater that someone added a heating element to. Just to reduce confusion:
1. Does any water come out when the hot water tap is on
2. Does it work/heat water on gas
sounds like what is being called a "Solid State" refrigerator
that uses a Pizeo electric cooling element. If so, don't expect much in the way of cooling. I tried one in the house and it wouldn't even get a 6 pack of coke cold after 12 hours. Took it back.
How do you know if the black tank is clean? You open the flush valve in the loo and look with a flashlight. There is also a water wand that is available at the RV stores that directs water down inside the tank for cleaning.
BTW: The ice bit works real good, but only when towing.