If the chime sounded in all positions but only when the key is inserted, I'd be inclined to check the door switch for monkey business or severe corrosion. Faults could exist in the courtesy lamp delay relay to cause this, and certainly the chime itself could be the cause of the behavior. Pull the cover from the blue box and see if it looks like water got in the electronics.

The cluster issues you bring up deserve another thread, but I wouldn't jump to the thermostat on that one. Even with a broken thermostat, you should see some gauge movement. Do as trichard suggests to ensure the yellow wire from the gauge sender is communicating with the gauge. You probably are looking toward the replacement of the odometer gear and excision of the temp compensator board.
I assume your speedometer functions normally and your fuel gauge works.

Some background on the above is here on my page: some reference pix
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Art Benstein near Baltimore

He who smiles in a crisis has found someone to blame.