If your in Sweden surely you want JAA as well? Unless you have the right to work in the USA. Why FLL? You can commute within an hour and a half to international airports pretty much anywhere in Florida
Your FAA licences are next to useless to you in Europe. i.e there are only a couple of jobs for FAA rated pilots in Europe - there are also hundreds of pilots after these - a much tighter job market than the JAA market.
There is no such licence as a frozen ATPL - what you actually mean is a JAA CPL with multi engine instrument rating - with your ATPL theories complete. What you think is a frozen ATPL is a JAA licence term not a FAA one.
Your in the wrong place for advice on FAA training - try
jetcareers.com for that.
34 is pretty average nowadays to be getting into training - especially now loans are harder to come by.
The first thing is you obviously need to read the sticky threads at the top of the professional pilot forums - your questioning style has come up many times before.