#2 The temp gauge is not controlled by the t-stat, its controlled by the temp sensor , sender , and/or computer
#3 Without knowing what engine we are working with (is it a 1967 327 or a 2006 6.0L) on older vehicles a stuck closed t-stat will cause the engine to get hot and if its stuck open it can cause choke issues with the carb (if you use it to warm up you engine) and the heater(if its winter). If its a newer (computer controlled) engine it can effect fuel economy, overheating issues and engine performance.
#4 As far as "How is the water pump driven?" What engine? , It could be either fan belt driven, timing belt driven or cam shaft driven.
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