Fuel Gauge
I had the same problem with the fuel gauge on my ski. My problem was the fuel float had sunk to the bottom of the baffel, the fuse was O.K. If it was me I would check the float first. The new design, which is far superior is one piece molded plastic, the old design used corks for floatation. And NO this would not cause a long beep.
As far as the cooling problem is concerned, I'm not sure what could be the problem. The cooling system is pretty simple, water flows from the jet pump, through the head of the motor and back out. The only thing I could think of is an obstruction, so check you lines, hook up and garden hose to check if water is flowing through the system. Are you sure its overheating?
Lou