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The deck probally has a bad heat sink on the output IC..... it overheats and the sound slowly goes out.
As to the ground, the negative post on the battery is the ground wire.... on a car the negative post is connected to the body which makes all the metal in the car a 'ground' on a boat the negative post is connected to a wiring harness that feeds ground to all of the electrical components..... thats why you don't get a ground from the metal in the console, I suppose you might have a bad ground..... to be sure I would hook the decks power and ground wires to a battery charger and see if it still stops working after 10 minutes, but I still think from your discription the output IC is bad.
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