Don't worry about giong to New York on your own. Even though I haven't been on my own, only with friends we have split up whilst there to do our own thing and I've felt very safe, certainly safer than whilst in London. Obviously you have to be careful as there are pickpockets everywhere, but as I say I felt safer there than other places.
The hotel I would recommend is the Milford Plaza. It is just one block from Times Square, and about 10-15 minutes walk from both Macys and The Rockerfeller Center. Even though other hotels may say that you have to show your hotel key to get into the lifts they don't (eg the Pennsylvania Hotel), yet at the Milford Plaza you ALWAYS have to show your card so you know that it is ONLY hotel guests and staff that have access to the hotel. You may be able to get a better deal on other websites but this is their website
http://www.milfordplaza.com/