1. The dorms have no AC. Fans are nice in the summer and early fall, but you can't have them during basic.
2. Don't worry about it. There will be plenty of time after basic to figure out your needs. Power cords and the like don't count as personal items. Also, they'll issue you a power strip.
3. I know people who used both. I had a non-electric and I always shaved at night. People with electric razors could shave in the morning and do it much quicker. The bathrooms in Jacks will have outlets, but sometimes they are very crowded. That being said, a good electric razor can last for quite a while on a single charge.