Testing the thermostat is relatively easy but you would remove it for placement in heated water. The temperature of the water would be monitored while observing how the thermostat opens and closes. You can read about test descriptions. It's just as simple (and pretty cheap) to replace the thermostat. After 20 years of service, I replaced mine. Temp gauge readings are the same before and after, but now I have a spare thermostat.
I'm not sure why you are questioning the operating temp. It doesn't sound like you monitored temp at the right location if you haven't monitored the thermostat housing yet. If you have any question about the coolant, by all means, replace it ... but I have my doubts that older coolant would affect the operating temp. You can run water through the system with the same results. The purpose of coolant (AKA antifreeze) is to prevent the liquid from freezing at extremely low temps, boiling at high temps and it prevents corrosion. Other than that, it doesn't affect the operating temp of the engine. If you aren't running the engine under load (driving through the water) you may not even be heating the engine fully simply by raising the rpms in neutral. But I suspect that you are simply reading coolant system temps in areas where the coolant is cooling down, not where the temp is highest. Like I said, the temp sender is not located where the circulating coolant is the most heated.