The FAA now refers to it as a "flight review". The term "BFR" is no longer used although it is required every 24 calendar months. If you take part in a 121/135 or military recurrent training program then the requirement is waived. However, this does not apply based on a foreign carrier.
You have a couple separate issues here. Doing a flight review or checkout at a flight school does not "convert" your Oz certificate to FAA certificate. I'm assuming you do not hold a FAA certificate hence your question.
The FAA will (as a courtesy) VALIDATE a foreign pilot certificate that will allow you to fly a "N" registered aircraft here in the states. This is not to be confused with CONVERTING your Oz cert to a FAA certificate.
If you want to come over and punch holes in the sky with a C172 then find a local FSDO in the area where you plan to rent the aircraft. You must contact the FSDO and they will tell you where the Oz CASA must send your pilot credentials. Verify the FAA has received your documents. This can take a while so plan in advance.
Contact the local flight school and complete a flight review with a CFI. This should suffice as a "check out" to rent the aircraft. Take your logbook endorsement, pilot certificate and medical certificate to the FSDO. They'll have you fill out a form and issue you a FAA validation.
It will say on the paper that it is issued based on your foreign pilot certificate subject to any restrictions imposed by your government along with medical requirements. It grants you private pilot VFR privileges only.
I don't recall any different limitations from Oz. However, pilots from say Mexico cannot fly VFR at night. Every FAA private pilot can fly VFR at night but if you "validated" your cert then you would still have the same restriction since you do not hold a FAA certificate. European pilots require a night endorsement. Without it, they would be restricted from flying at night as well.
Converting your foreign pilot certificate requires you to take the required FAA written exam, obtaining a FAA medical, and taking a checkride with a FAA examiner. You will have to document the required experience (which you likely have in your logbook) as well for each rating.