No, it doesn't. In either direction ie FAA FR count for Oz nor Oz count for FAA (presuming the instructor isn't rated in both countries). See my 61.75 comment below
re TSA checks: It's not just the first FAA licences that requires TSA approval, but any licence or rating that increases one's ability to fly something bigger, faster &/or heavier. Private, Commercial, Instrument & every type rating *all* require TSA background checks.
re 61.75: Valid only as long nothing changes on the host licence ie licence number, country of issuance etc. If anything like that changes then a new 61.75 needs to be issued. The flight review requirement comes from the requirement, elsewhere in FAR 61, for a 24 month review. The rules specify who may give a flight review. That 'who' is an authorised instructor - and the only authorised instructors are those with locally issued instructor ratings.
Last edited by Tinstaafl; 18th Jun 2013 at 00:59.