Need your advise on studying RN in NY/NJ

Durian, I've never heard of a pre-RN exam. If your grades in the prereqs were Bs or better you should have no trouble getting into a two-year RN program at a community college.

I have no idea about licensure.

It is not so much passing the pre-RN or entrance exam, that is the issue or that I would be concerned about, especially if you already have completed your pre-reqs (chem, bios, anatomies). Here in NJ, there is a waiting list for many of the schools (2Y ADN). I know that Raritan Valley CC and UMDNJ (MIddlesex) both had long waiting lists. Not too sure about Muhlenberg, they may have a waiting list too now, not sure. I know all 3 have an entrance exam, which is like SATs with a math/science/english component. If you already have most of the pre-reqs, I would suggest a 4Y program, less of a chance that you'll have to wait before you can start taking your Nursing curriculum. Just a suggestion.

I know if you are registered (RN) in one state, to work in another state, you need to check with that state you want to work in and see if there are any other things you need to do prior to transferring over your license to them (ie, add'l CEUs, etc). Although, I would not worry about that until after I have finished school and passed the NCLEX, but that's just me.
Good luck to you.

After graduating from your Nursing school, you can apply for initial licensure in any of the 50 states and the Disctrict of Columbia. You should not have any problem. But each states has a different requirement-for instance, Ny requires that you complete courses on Child abuse mandatory reporting and Infection control before being given a license.