I often sleep with one or two of my bedroom windows open. I hate waking up to a stuffy room that has no fresh air.
But I've noticed that if I sleep with the windows open, and the temperature drops to an uncomfortable level during the night, I will either; have a nightmare which wakes me up, have an unpleasant dream that will wake me up. I will then close the window and sleep much better, with no recurrance of bad dreams.
Doesn't get super cold where I am, but I'm the same about wanting fresh air and disliking "stuffiness." If bedroom temp is over 62F (66F is "tolerable") I definitely don't sleep as well. Doesn't affect my dreaming that I've noticed, however. But again...not very cold here. Coldest I've been in was in brother in law's house further North, in snow, sleeping in a non-heated basemant under piles of blankets and long underwear. Must've been 35F in that basemant, with windows shut.
In some of the northern parts and in the deserts, yes, but I live 15 minutes from the beach (good traffic). Moderates the temperature too much. I don't think it's ever been below 5 degrees Celsius here.
And yet everyone around here still complains about it being cold and runs the heater all winter...
Heh. Yup, we're often cold-wimps. My Canadian friends laugh at me when I complain that 37F is "too cold".

By SF Bay, we get near-freezing temps for 2-3 week (total) semi-frequently in winter, but that's about it. Winter's are mostly low-high 40's...sometimes low 50's. Almost tropical!