If, in this weather, you tend to have the cabin heating turned up, what you describe is quite normal: the engine only makes a certain amount of excess heat, and you use most of that heating the cabin.
And, to your original question: Usually, no. Well, you can make almost anything fit with a hammer, but the usual arrangement is that the thermostat will not fit unless it is the right way around. And, even if you did fit it the wrong way around, it would probably work more-or-less normally unless you are giving it some stick, and then it might overheat. So, it doesn't seem likely as a cause, particularly as you seem to be saying that it hasn't been touched.
- Phil1 likes this