Yes it would be a new warranty because the phone is new. The Storm warranty would continue to have as much time that is left on it. You may have to transfer the insurance to the new phone but that should trake care of it. Just talk to customer care and explain to them they should tell you what the best thing to do would be.
As long as you keep Verizon service, I really don't think they care if you switch devices (or how often). I would say just talk to Customer Service, and they will help you out.
VZW insurance is non-transferable. Once the phone is off the line that originally had it, it is no longer covered. Trust me, I need to explain this to atleast one person a day. The warranty will be covered, but not insurance.