You can only do a JAA IR in a JAA state, not in the USA. You can fly an N reg anywhere in the world on an FAA license. You can fly a JAA reg anywhere in the world on a JAA license (with a few small caveats). Individual states can choose how they validate other states' licenses, and the rules vary. JAA licenses are easily converted to FAA licenses (PPLs, at least), and FAA licenses are valid to fly most JAA register aircraft in JAA states. Ratings are a little different, and more complicated. Converting 1 license to another doesn't mean you lose a license, it means you have both.
Whilst slightly abrasive, KeithTo is absolutely correct, the search function will tell you everything you need to know. I use it all the time and it always turns up what I need. If your username is indicative of your age you would appear to be one of the "had internet before I had pubes" generation, in which case I'm surprised you can't get satisfactory results from the search facility.
Reading LASORS and FAR/AIM is a good start.