I wouldn't bother cutting your hair before you go, unless you want to. they cut it hideously short (about 3 inches on top and 1 inch in the back). it's alright though, all the females look the same. you will stick out, however, there are definitely more baldies than people with hair. i remember the first week one of the guys in our class thought "why do those GUYS get to keep their hair?" apparently not the brightest bunch, but it turned out ok.
as far as running goes, they do have some specialized group runs. this means that you will be put into a group that runs similarly to you. for example, those who run an 11 minute mile will be in the 10-11 group. it's not divided by sex and in fact they didn't happen very often in my experience. there are long runs (max. 5 miles, average: 2.5miles) in formation (as a class or group of classes) so be prepared to shorten your stride.
as far as products are concerned, all those things you mentioned are acceptable. you will be allowed a toiletries bag that can contain everything you need to feel as comfortable as possible (which isn't much). no meds though. self-medication is definitely not acceptable when you are there.
staying under the radar is impossible for females. you automatically stick out due to the hair factor, so prepare yourself for the unwanted attention. one way to lessen this is to make sure you can PT well. work as hard as you can to meet the guys' standards. my roommate and i did this in our class and our drill instructor left us alone.
let me know if there are more questions