I am having a broody chicken issue, too. I just put her in cold water for about 5 minutes, then put her back in the pen, but blocked off the coop and nesting box.
She went over and drank quite a bit of water, then started scratching around. Last I checked, she was giving herself a dust bath...
I have been wondering the same thing about cold temps and my 6 week old leghorns and golden comets. It has been in the 50's and a cold rain...they huddle together in a corner under their coop. Question...when will they learn to go into their coop to get out of the chill? or maybe they won't...
My chickens spent their first night out in the coop on Saturday. I was so nervous! At dusk, I had to put them in the coop and shut the door. They peeped for a while ...then I waited a half hour and checked on them...they were all quiet and snoozing
They spent the next day out in their pen...