IT's CUTE! BUT for that price, I can build two (or more if utilize free, inexpensive or less expensive supplies from Habitat stores, Craigslist & FB Marketplace) 8x8' square by 6' tall CP (cattle panel) hooped coops that will house a lot more birds and I can walk into standing up w/o a problem. Or build a solid, enclosed coop in the back portion, and add extra hoops to make your run however long you want to. Several plans in the coop section show how to do these w/ different people doing different frames for front and back. Different roofing options, different framing options, even different ways to do the base if you want it stationary (on wood or directly on the ground) or movable (tractor - w/ or w/o wheels). Sooo many options and easy to build, you can put most, if not all, of it together by yourself. The biggest expense then is the hardware cloth to cover the CPs and the tools that you'll need to build it (if you don't already have them).
They would work well in your hot/humid climate (it was 100* at the high today w/ a 120* heat index here) w/ so much air flow. They will flex and hold during hurricanes - some tarp roofs will remain, some will not (there are many different roof options). You mentioned a forested area - if you only cover/roof/enclose a smaller cooped area, the CPs should even withstand limbs, branches landing on it w/o damage (again, ours have - just needed to replace the less expensive tarps that didn't survive the wind storm or hurricanes). During the colder portion of the winter or during the couple of ice/snow storms, you can put up extra tarp, plastic sheeting or shower curtains as someone else already stated. The weight of ice during the last 4-6 storms hasn't damaged our hooped coops at all - no collapsing, snow came off (not a lot of heavy/wet snow) w/o a broom, but could easily have broomed off the snow if needed.
The shed is a nice idea, too, and later if you don't keep chickens and/or you sell your property, you can convert it back to a regular shed. There are many ways to build a safe run off of the shed for your new girl (and the new birds that will now find you).
