I'm going to take a big Sherlock Holmes guess based on "clues" I think I've read:
Clues:
You said each hen shows up with a clean cut, bleeding, on top of foot. Then goes to roost and won't come down and later dies. Prior to that they are seen crowding in front of the fan for cool air.
Is it possible the fan is cutting their feet? If not, something is or they are stepping on one another's feet and injuiring them in their efforts to get to the cool air. Once they've been injured, I think it scares them and they get back on the high roost and won't come down. Once up there, in the heat of the day, they die of heat stroke. Hot air rises - plus if they stay up there any length of time during the heat of the day, they aren't getting enough cool water.
I would watch them while they are in front of fan and see what takes place. I don't think a rat is running into a group of chickens and biting the top of their feet. It wouldn't leave a nice clean cut anyway. I think they are cutting each other's feet by crowding or fighting to get in front of fan - or the fan is cutting their feet.
We have an old hen house with ceiling roosts about 8 feet up. The chickens go up there at night. But during the heat of the day they are all on the floor of the coop - often crowded together, in front of the fan. Their coop stays reasonably cool because it has a high vaulted ceiling and lots of ventilation but they still stay on the floor - often digging holes and laying flat out, face down, in the cool dirt floor.
If it's hot where you are and they are trying to get cool but then something hurts them and they go and stay where they are "safe" like a high roost - the heat up there will kill them quickly espcially if they don't come down for water.
Try moving the fan where they can get the breeze easier - or adding another fan. We bolted our fan to the wall and angled it down to the ground but it is also covered with a plastic cage so they can't hurt themselves. Also, try putting shallow pans of water for them to stand in or leave the water hose running on low in their run so the ground floods and they can get in the water and wet dirt - this is my chickens and ducks favorite thing to do. Mine all freerange our farm but with temps at 100 degrees it's hard for even them to stay cool under bushes and such, I can't imagine what cooped up chickens must be enduring. We have water hoses running on low all over our property and that's where everyone congregates.