I thought the fan was definitely a good item to bring. The rooms are even more humid than it is outside, and you'll learn that it's impossible to dry any clothing you have with the air alone. The fan can be used as a way to dry yourself and your clothing, especially after the mock Sea Trials. Trust me, your mom will not be very happy to find wet clothing in your suitcase, no matter how hard you worked to dirty and wet that clothing.
To address the "time of the month" issue: You'll have time to do what you need to do, especially during the workshops. The professors are extremely nice and will happily tell you where to find a bathroom, even during the middle of the activity (from personal experience from hydrating). The only suggestion I really have is to pack something in your towel roll for the mock Sea Trials. You won't really have time to go to the bathroom during those six hours other than when you change into your bathing suit for the swimming evolution because you're always on the go that day. So again, just stick something in your towel roll, bring a rubber band or something to hold the towel, your bathing suit, and feminine product in, and you'll be good to go!
Oh, and a large majority of my platoon applied to USNA or another service academy, and most of them got in. Out of my squad, including myself 5/7 got an appt. and one other got an appt. to USAFA.
If you guys have any more questions, feel free to shoot me a PM.
GO NAVY, BEAT ARMY!