Hey,
Fellow Floridian here. I've had my flock since March and have a similar open air set up with ¼" hardware cloth & a tin roof. I also have a little solid plywood alcove for the nesting boxes. They seem to like it a lot.
I'll put up tarps to block drafts if there are hard freezes in the winter, but I'm more concerned with keeping them cool.
I also have a shade cloth over their run and lots of waterers & a box fan.
The only problem I see with putting your coop directly under trees is if there are big branches that could fall down during a storm & damage the coop?