Any school that can do the JAA PPL in the US will also be able to do the FAA licences. Getting your visa transferred is not straightforward.
I can't quite see why you want to do an FAA ATP if your aim is to obtain a JAA fATPL. Unless you're already an airline pilot, the only FAA qualification that has any real conversion value is the IR. Even the Commercial isn't really worth the cost and hassle of obtaining it.
Regarding the M-1 and J-1, both are multi-entry so you shouldn't have any problems in that regard. There are not many schools that can issue the J-1 and it is meant for people who wish to hour build as an instructor, normally for up to two years, to obtain an FAA ATP. This does not convert to a JAA ATPL though unless, as I have already said, you have a reasonable amount of airline experience. You will only get the same terms as someone with a Multi-Engine Commercial and Instrument.