Going back and forth over winter break I definitely noticed a productivity shift being some place I was warm and comfortable.
Perhaps there are less noticeable psychology benefits for it being cold but I do not think being uncomfortable and dealing with it is a long term strategy for success. And probably completely impractical if you live with other people in the same house.
If that isn't sufficient, there are still cheaper ways to deal with it than adjusting the furnace. A low power radiative space heater (the sort with a reflective dish) can be very efficient at heating up the person it's pointed at without heating up the whole room. Having a warm drink to pick up and sip on also helps; it'll warm your fingers directly and also give your core a little boost of heat. Heated electric blankets are also extremely effective.
Bathrooms and kitchens are a special case, which can be addressed with ventilation. One of the upsides of keeping your home cool in the winter is it's much cheaper to circulate in new air from outside. You get to have fresh air all winter without paying to heat all that air up. I believe fresh air throughout the year has many health benefits.
It might be like you said only if you don't use that room, or you have a big house so maybe you don't use most rooms and then you don't fill them with humidity from your own breath. Walls are cold but also dry.
Opening the windows in the winter is how you let all that humidity out.
Absolutely, being able to open your windows is a big advantage of keeping your thermostat low.
I find that in cold rooms, my fingers slow down even if I'm warmly dressed.
typing with cold fingers might be a recipe for arthritis.