When you replace the thermostat, use the electronic odometer trick to monitor the thermostat operation while driving. You will have to familiarize yourself with temps in C, but you should see the coolant heating up to the temp of the thermostat, then as the thermostat opens coolant temp will drop rapidly, and then begin to heat up again. Eventually it will settle above the thermostat temp unless the ambient temperatures are extremely low.
Key off, press and hold the odometer reset button, turn the key on, hold the odo button until the odometer reads [test], release and it will immediately change the [gage] and the gauge needles will jump around. After 3 secs that test is done, and each time you press and release the odo reset button you will scroll through one of 30 test functions. Most of which will mean nothing. Look for the one, about 10 presses, that reads like this [15 C] that is assuming an ambient temperature around 60F, and a cold engine. Your reading will probably be different, but that is the format. Now, without turning the key off, start the engine. The odometer should remain in test mode, and you should see the temp rising within seconds. Drive, as driving is the best way to warm up an engine and take it through the cycle.
Normal thermostat operation will be as listed above, cycling up to the thermostat temp, then down, back up again until it settles just above the thermostat temp.
A bad thermostat will not cause you to lose coolant, so that is not going to solve your problem. You have a leak somewhere. I had a problem with a friend's car that took me a long time to figure out. I only figured it out after several short drives, and then sitting idling. She had a bad coolant cap. When the engine heated up, it would release coolant through the overflow while she was driving. The symptoms were the same as yours= losing coolant, no steam or coolant smell in exhaust, no coolant in oil, etc. I tried to diagnose it by inserting a Q-tip in the end of the overflow. I figured that if the Q-tip was blown out, then it was a problem with the cap. I was never satisfied that the Q-tip was being pushed out by coolant and not simply falling out due to wind while the car was driving. After a few short drives, the overflow peed out coolant at idle while I watched verifying the problem. Car engines will heat up immediately after stopping driving for a few seconds, it will really heat up if you turn off the engine immediately after applying a load to the engine.
