1. Boil a pot of water on your stove, now drop the thermostat into that water and you should be able to see the thermostat open up. If the thermostat doesn't open than it is bad and needs to be replaced.
2. Yes you can replace the thermostat on your own, it's really quite simple. Make sure when you buy a new thermostat that it comes with a seal because this eliminates the need to buy a separate gasket. Before installing a new thermostat, place the new thermostat in boiling water so it can open up for the first time. Sometimes if you don't do that than your car will overheat before the thermostat opens up for the first time. To locate where the thermostat is on your vehicle, follow the upper radiator hose to where it meets the engine. That is where the thermostat is located. You didn't mention what engine you have but you may need to remove something to gain better access. Now drain your coolant into a clean container because if your coolant is in good shape you can reuse it and you don't want other contaminants in your coolant. Now just remove the thermostat housing which usually just requires removing a couple bolts and pull the housing away from the engine. Remove the thermostat and clean the surfaces of the housing and the engine so you get proper seal. Install new thermostat in the same way it came out, DON'T INSTALL IT BACKWARDS!!! Now put everything back together and refill your cooling system with either new coolant or reuse your drained coolant if it is still good. Run your car for fifteen minutes, make sure it doesn't overheat, check for leaks and top off coolant as needed.
Good Luck!