What happens to a thermostat that it will not open?
Thermostats, like other parts, just fail. That's why there are about a zillion of them for sale in the auto parts stores. Just pick up a new one, if you think yours has failed. FriPilot
Yes they can stick open or closed. An automobile thermostat is just as likely to stick open as it is to stick shut. Symptoms of a stuck open thermostat are slow to reach normal operating temperature and only warm air blowing from the heater. Symptoms of a stuck closed thermostat are engine over-heating and radiator boil-over.