I live in Sydney, Australia, where I fly my "N" registered Mooney on my FAA license. I've had no problems (from CASA or otherwise) operating this way.
My question concerns IFR flight and FAR 91.411 - Altimeter system and altitude reporting equipment tests and inspections, FAR 91.413 - ATC transponder tests and inspections, and FAR 43 requirements.
Is it true that 91.411 and 413 applies only to flights in US controlled airspace, and if so, is my compliance then with Australian regulations?
ANSWER: Yes, FAR 91.411 and 413 apply to operations within the United States under IFR. Yes, the rules of the country you are in would apply for operation of the aircraft.
Does that mean that I can have a valid logbook entry for a FAR 43 Appendix E or F check done by an Australian avionics shop as long as they comply with CASA regulations for the certification?
ANSWER: No, the rules of the country of registration (United States in your case) would apply for aircraft maintenance and airworthiness certification.
Who may legally perform this check on an N registered plane outside the USA, and is it legal to put that entry in my logbook?
ANSWER: When any maintenance is performed on your "N" registered aircraft, a licensed and qualified person must perform the work and make the record entries. That means only a U.S. certificated mechanic or a U.S. certificated repair station can perform the work.