We bought our son a cheap watch to take with him, just in case they didn't let him keep it. It was a regular black digital watch. I plan to take his good watch to him at PIR. He hasn't told me whether they let him keep the cheap one or not.
Yes he can. Check out: http://www.bootcamp.navy.mil/what_to_bring.asp
The following is a list of authorized personal items that you may want to bring to training, but are not required. Any unauthorized or unnecessary items (including your civilian clothes) will be boxed up and shipped home at your expense.
Note: Female recruits are not authorized to bring a swimsuit, underwear or bras. All will be issued upon arrival.