Hello: A thermostat is likely bad, though it could be an element with a leak to ground (see tanklets). The lower thermostat is more likely to fail than the upper one as it gets more use and is more likely to get wet. So, test elements for leakage to ground and see if the lower thermostat can be shut off once the tank is hot.
About that valve leak, you prolly have a gate valve that’s leaking from the valve stem. If so, you can loosen the packing nut, wrap the stem in lots of teflon and snug up the packing nut. I’ve had good luck with that approach.
Heaters that haven’t been paid attention for twenty years tend to take out their frustration on the nearest human 😯