If you're not using coolant I wouldn't be concerned about a head gasket.
Yes, you may have problem with a stuck thermostat or a missing one. I know 350 is missing one, will remedy that this winter.
As for and abnormally strong water pump, what's your idle speed? If memory serves should be about 425 rpm.
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128 X 2, 129, 149, 1250, 1450 x 2
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