yes you can bring a small portable fan packed in your luggage, lots of folks do it.
The partitions between balconies are locked.
you can prop the balcony door open or hold it with a bungee. Just remember to shut your ac off if you have the door open.
drink the ships water, it is purified and filtered, better than the bottled water you can buy. Bring aboard a few bottles of bottled water when you board and just refill from your cabin.
If you haven't experienced the StingRays in Grand Caymen its a must do, if you have done it and don't want to do again you can go ashore for awhile and shop the main street, wander around and still do your spa time aboard the ship