Here's a good place to start!!
However, a quick answer to your questions:- Yes, quite a few people train in the US and convert to JAA licence. However, it is not always cheaper in the long run and may not be as straightforward as some schools make it out to be! Secondly, in general, JAA aircraft cannot be flown on an FAA licence (there some exceptions but special authorizations are required and are rare) so, if you want to work in Europe (because you are only entitled to work in Europe for example), then you really need to have a JAA licence.
Cheers
Whirls