For what I know in Emirates the medical is part of the selection and is taken during the days of the interview.
In Qatar is different, once you start working there you do the medical, if you do not pass it, you go back home... It is done during the first two months, while you still are having classes and dealing with the burocracy.
in Saudi Arabia they work with the faa rules, and the faa license I believe is accepted... So no need for conversion. Just a thought, I am not sure about it.