I let mine sit in the nesting box, no eggs, but take them out a couple times a day and put them in the run so they will eat and drink. They stay out for awhile and then head back to the nesting box. We pick up all the eggs, not hatching anything. Mine are all bantams and maybe not as grouchy as some broodies, and easy to handle. I take them out and give them a little special attention/treats and wouldn't dream of dunking them in a bucket of water or putting ice under them, but yes, they are pets.
My little Cochin frizzle, Esmerelda, is broody and this week has decided to move from one box to another, wherever she can find some eggs. Last week she just sat in an empty box.
deb g