Do you have to drain the coolant to replace the thermostat?

You will have to drain part of the coolant. Try to drain as little as possible and fill up before restarting the engine.

No, the thermostat is usually at the top of the engine block (placed between the top input hose and the block). If, however, the radiator is full of coolant, there will be some spillage. Also, make sure the coolant has cooled down before replacement, and do periodically replace the coolant as needed.