Is breaking a FAR an issue when flying outside of FAA jurisdiction? When flying outside the US it is advisable to follow the FAA rules when they are stricter than ICAO, and when it does not violate the local flight governing body's rules.

Can you still be fined or get into trouble or does the FAA only have jurisdiction in USA airspace?
The FAA only has Jurisdiction over US airspace and you must follow the rules of the airspace you are flying in/through. If you are talking about N registered airplanes they must follow FAA rules for airworthiness etc. even if they are kept outside the country.  

Here is in interesting question on breaking laws on a plane in general (sort of applicable here)

Side Note: Although the FAA does not have jurisdiction outside US airspace if there is an accident involving a US built aircraft (mainly Boeings at this point) the NTSB will send a representative to compile their own report (and aid in the investigation). I know this happens for commercial accidents I am unaware if they send representatives for US build GA plane accidents outside the country. 

Side Note 2: US pilots licenses are valid elsewhere in the world and allow you to fly elsewhere in the world certain actions elsewhere in the world (if caught/reported) may lead to the revocation of your US issued license. There is some info on that here