I hold both an FAA and JAA MEP. I have flown on my FAA MEP here in Ireland in an N Reg a/c. To fly under a FAA license in JAA land (so to speak) you need to make sure to keep your FAA medical current even though you may hold a JAA class 1.
I did the FAA MEP first after my FAA PPL, then did the full JAA CPL/MEP. I didn't convert. I did my JAA IR after that so I can't really answer some of your questions there sorry. I don't hold an FAA IR. I think offhand the conversion hours are small enough from FAA -JAA for the MEIR. I think around 5 hrs in min requirement in a/c. I know my school used to have a conversion program for MEIR and that was 5 hrs in PA44, 10 hrs in FNTPT2 and then ground school.
I personally found the FAA MEP license much better. More dept to it. Really goes into Vmc demos and feathering props. In my school in Ireland we never did an actual Full engine shut down in air (Did one everyday in US during MEP). That may have been the schools rules or the IAA I'm not too sure. The FAA course I did was 15 hrs in a/c and 5 in FNPT2 sim (I think) and the JAA was around 8 hours for the MEP section. It's been few years since I did them so I may have the hours wrong.
So would highly recommend doing the FAA, really enjoyed it and learned a lot.