It's hard to follow along here what is going on.
So, your stove and water heater work fine on gas, with shore power connected or disconnected, right?
But your fridge and furnace both audibly spark, but do not ignite, with shore power connected or disconnected?
Both your furnace and refrigerator may have a gas valve at the appliance. Both of mine do. Make sure they are both on.
Clean the ignitor electrodes. Have a helper turn on the appliance while you watch the electrodes. You should be able to see them sparking. You can also try to use one of those long-neck lighters to manually light the flame when someone turns on the appliance. If the spark is not doing the trick the lighter will - if you have gas flowing.
If you can get the appliance lit with a lighter, the problem is with your ignition system. It is a simple control board. Clean all contacts and the ignitor electrodes. If this fails, you may need new control boards. I just replaced my water heater and refrigerator control boards with ones from Dinosaur Electronics. They work great.
Steve