Can I remove the droppings tray of the coop to cool it down at night?

We would not recommend doing this as predators can gain access to the coop through the roosting bars.

For heat we would advise to put ice in the droppings trays, small fans in the coop (such as battery clip on ones), a gazebo over the coop to keep it in the shade, melons inside for the hens, and water sprinklers.